Introduction To Mathematical Analysis-Igor Kriz
Introduction To Mathematical Analysis-Igor Kriz
Introduction To Mathematical Analysis-Igor Kriz
com
Igor Kriz
Aleš Pultr
Introduction to
Mathematical
Analysis
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Igor Kriz
Aleš Pultr
Introduction
to Mathematical
Analysis
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To Sophie
To Jitka
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Preface
vii
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viii Preface
Because of this, the aim of our book is not limited to undergraduate students. This
text may equally well serve a graduate student or a mathematician at any career
stage who would like a quick source or reference on basic topics of analysis. A
scientist (for example in physics or chemistry) who may have always been using
analysis in their work, can use this book to go back and fill in the rigorous details
and mathematical foundations. Finally, an instructor of analysis, even if not using
this book as a textbook, may want to use it as a reference for those pesky proofs
which usually get skipped in most courses: we do quite a few of them.
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Contents
1 Preliminaries . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1 Real and complex numbers .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Convergent and Cauchy sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 Continuous functions . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4 Derivatives and the Mean Value Theorem .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5 Uniform convergence .. . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
6 Series. Series of functions .. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
7 Power series . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
8 A few facts about the Riemann integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
9 Exercises . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2 Metric and Topological Spaces I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1 Basics. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2 Subspaces and products . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3 Some topological concepts . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4 First remarks on topology . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5 Connected spaces . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6 Compact metric spaces . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
7 Completeness . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
8 Uniform convergence of sequences of functions.
Application: Tietze’s Theorems .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
9 Exercises . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3 Multivariable Differential Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
1 Real and vector functions of several variables . . . . . . . . . . . . . . . . . . . . . . . . . 65
2 Partial derivatives. Defining the existence of a total differential . . . . . . 66
3 Composition of functions and the chain rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4 Partial derivatives of higher order. Interchangeability . . . . . . . . . . . . . . . . . 74
5 The Implicit Functions Theorem I: The case of a single equation . . . . 77
6 The Implicit Functions Theorem II: The case of several equations . . . 81
7 An easy application: regular mappings and the Inverse
Function Theorem . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
ix
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x Contents
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Contents xi
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xii Contents
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Contents xiii
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505
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Introduction
The main purpose of this introduction is to tell the reader what to expect while
reading this book, and to give advice on how to read it. We assume the reader to be
acquainted with the basics of differential and integral calculus in one variable, as
traditionally covered in the first year of study. Nevertheless, we include, for the
reader’s convenience, in Chapter 1, a few pivotal theoretical points of analysis
in one variable: continuity, derivatives, convergence of sequences and series of
functions, the Mean Value Theorem, Taylor expansion, and the single-valued
Riemann integral. The purpose of including this material is two-fold. First, we
would like this text to be as self-contained as possible: we wish to spare the reader a
tedious search, in another text, for an elementary fact he or she may have forgotten.
The second, and perhaps more important reason, is to focus attention on facts
of elementary differential and integral calculus that have deeper aspects, and are
fundamental to more advanced topics. In connection with this, we also review in the
exercises to Chapter 1 definitions of elementary functions and proofs, from the first
principles, of their properties needed later. What we omit at this stage is a proof of
the existence of real numbers; the reader probably knows it from elsewhere, but if
not there will be an opportunity to come back and do it as an exercise to Chapter 9.
An entirely different prerequisite is linear algebra. While not a part of mathemat-
ical analysis in the narrowest sense, it contains many necessary techniques. In fact,
differential calculus (in particular in more than one variable) can be without much
exaggeration understood as the study of linear approximations of more general
mappings, and a basic knowledge in dealing with the linear case is indispensable.
The reader’s skills in these topics (determinants, linear equations, operations with
matrices, and others) may determine to a considerable degree his or her success with
a large part of this book. Because of this, we feel it is appropriate to include linear
algebra in this text as a reference. In order not to slow down the narrative, we do
so in two appendices: Appendix A for more theoretical topics such as vector spaces
and linear mappings, and Appendix B for more computational questions regarding
matrices, culminating with a treatment of the Jordan canonical form.
Let us turn to the main body of this book. It is divided into two parts. One
of our main goals is to present a rigorous treatment of the traditional topics
of advanced calculus: multivariable differentiation, (Lebesgue) integration, and
differential equations. All this is covered by Part I, including basic facts about line
integrals and Green’s Theorem.
xv
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xvi Introduction
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Introduction xvii
introduce Borel sets and prove criteria of measurability. We present a rigorous proof
of a multvariable substitution theorem. Finally, we introduce Lp -spaces: while this
may seem like an early place, we will have enough integration theory at this point to
do so, and to prove their basic properties. This is useful, as the Lp -spaces often occur
throughout analysis (for example, in this book, we will use them in Chapters 13
and 15 in proving the existence of a complex structure on an oriented surface with
a Riemann metric.) We will return to the study of Lp -spaces in Chapter 16, where
they provide the most basic examples in functional analysis.
Next, having covered differentiation and integration, we turn to differential
equations. We restrict our attention to the ordinary differential equations (ODEs),
as partial differential equations have quite a different flavor and constitute a vast
field of their own, far beyond a general course in analysis (even an advanced one).
For a text on partial differential equations, we refer the reader, for example, to [5].
Chapter 6 on (general) ordinary differential equations is in fact independent of
Chapters 4 and 5 and uses only the material of Chapters 1, 2 and 3. We introduce
the concept of a Lipschitz function and prove the local existence and uniqueness
theorem for the systems of ODEs (the Picard-Lindelöf Theorem). We also discuss
stability of solutions and differentiation with respect to parameters. Further, we
discuss the basic method for separation of variables, and finally discuss global
and infinitesimal symmetries of systems of ODEs (thus motivating further study of
vector fields); also, we explain how the methods of separation of variables discussed
earlier are related to symmetries of the system.
Chapter 7 covers some aspects of linear differential equations (LDEs). The global
existence theorem is proved, and the affine set of solutions of a linear system is
discussed. We show how to use the Wronskian for recognizing a fundamental system
( basis) of the space of solutions of a homogeneous system of LDEs, and how to
get solutions of a non-homogeneous system from the homogeneous one using the
variation of constants. Also, we present a method of solving systems of LDEs with
constant coefficients, easier in the case of a single higher order LDE, and requiring
the Jordan canonical form of a matrix from Appendix B in the harder general case.
Chapter 8, concluding Part I, treats parametric curves, line integrals of the first
and second kind and the complex line integral. At the end we prove Green’s
Theorem, which we will need when dealing with complex derivatives, but which
is also an elementary warm-up for the general Stokes’s Theorem.
Part II. Now our perspective changes. The traditional items of advanced calculus
have been mostly covered and we turn to topics interesting from the point of view
of geometry.
To proceed, perhaps by now not surprisingly, we need another installment of
topological foundations. This is done in Chapter 9, presenting more material on
topological spaces (separability, compactness, separation axioms and the Urysohn
Theorem) as well as on metric spaces (completion, Baire’s Category Theorem). In
the last section we prove the Stone-Weierstrass Theorem providing a remarkably
general method to obtain useful dense sets in spaces of functions, and the Arzelà-
Ascoli Theorem, which greatly clarifies the meaning of uniform convergence, and
will be useful in Chapter 10 when proving the Riemann Mapping Theorem.
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xviii Introduction
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Introduction xix
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xx Introduction
reading. There is no need to cover entire chapters, there are endless possibilities how
to mix and match topics to create an interesting course.
The student (or reader) is, in any case, most strongly encouraged to keep the
book for further study. As already mentioned, we anticipate that graduate students
of mathematics, mathematicians and scientists in areas using analysis, as well as
instructors of courses in analysis will find this book useful as a reference, and will
find their own ways through the topics.
In the Bibliography section, the reader will find suggestions for further reading.
In the more advanced sections of this text, we often introduce concepts (such as
“Lie group” or “de Rham cohomology”) which arise as a natural culmination of
our discussion, but whose systematic development is beyond the scope of this book.
These concepts are meant to motivate further study. We would like to emphasize
that our list of literature is by no means meant to be complete. The books we do
suggest all have a fairly close connection to the present text, and to mathematical
analysis. They contain more detailed information, as well as suggestions of further
literature.
Finally, we would like to say a few words about sources. The overall conception
of the book is original: we designed the logic of the interdependence of topics, and
the strategy for their presentation. Many proofs are, in fact, also “original” in the
sense that we made up our own arguments to fit best the particular stage of the
presentation (the book contains no new mathematical results). Given the scope of
the project, however, we did, in some cases, consult lecture notes, other books and
occasionally even research papers for particular proofs. All the books used are listed
in the Bibliography at the end of the book. In the case of research papers, we give
the name of who we believe is the original author of the proof, but do not include
explicit journal references, as we feel an effort of being even partially fair would lead
to a web of references which would only bewilder a first-time student of the subject.
We did want to mention, however, that there are also quite a few proofs which seem
to have become “standard” in this field (including, sometimes, particular notation),
and whose original author we were not able to track down. We would like to thank
all of those, who, by inventing those proofs, contributed to this book implicitly. We
would also like to thank colleagues and students who read parts of our book, and
gave us valuable comments. Last but not least, the authors gratefully acknowledge
the support of CE-ITI of Charles University, the Michigan Center for Theoretical
Physics and the NSF.
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Part I
A Rigorous Approach to Advanced Calculus
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Preliminaries
1
The typical reader of this text will have had a rigorous “ı-"” first year calculus
course, using a text such as for example [22]. Such a course will have included
definitions and basic properties of the standard elementary functions (polynomials,
rational functions, exponentials and logarithms, trigonometric and cyclometric
functions), the concept of continuity of a real function and the fact that continuity
is preserved under standard constructions (sum, product, composition, etc.), and the
basic rules of computing derivatives. We review here mainly the more theoretical
aspects of these topics. The reason for reviewing them are two-fold. The first reason
is that we would like this text to be as self-contained as possible. The second reason
is that some of the basic results have, in fact, substantial depth in them, and the more
advanced topics on which this book focuses make heavy use of them. Not reviewing
such topics would at times even create a danger of circular arguments.
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4 1 Preliminaries
1.1
Let us summarize the structure of the set R of real numbers as it will be used in this
text. We do not give a rigorous construction of the real numbers at this point. Such a
construction however will emerge in the context of our discussion of completeness
in Chapter 9, where it is reviewed as an exercise.
First R is a field, that is, there are binary operations, addition C and multiplica-
tion (which will be often indicated simply by juxtaposition) that are associative
(that is, a C .b C c/ D .a C b/ C c and a.bc/ D .ab/c) and commutative
(that is, a C b D b C a and ab D ba) and related by the distributivity law
(a.b Cc/ D abCac). There are neutral elements, zero 0 and one (also called unit) 1,
such that a C 0 D a and a 1 D a. With each a 2 R we have associated an element
a 2 R such that a C .a/ D 0; almost the same holds for the multiplication where
we have for every non-zero a an element a1 (also denoted by a1 ) such that a a1 D 1.
Furthermore there is a linear order on R (a binary relation such that a a,
that a b and b a implies a D b, that a b and b c implies a c, and
finally that for any a; b either a b or a D b or a b), and this order is preserved
by addition and by multiplication by elements that are 0.
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1 Real and complex numbers 5
is the least upper bound (resp. greatest lower bound), if it exists. Thus, the supremum
s of M is characterized by the properties
(1) 8x 2 M; x s, and
(2) if x a for all x 2 M then s a
(similarly for infimum with instead of ). (2) It is often expediently replaced by
(2’) if a < s then there is an x 2 M such that a < x
(realize that (1)&(2) is indeed equivalent to (1)&(2’)). It is a specific property of the
ordered field R that
each non-empty M R that has an upper bound has a supremum
or, equivalently, that
each non-empty M R that has a lower bound has an infimum.
In mathematical analysis, it is often customary to use the symbols 1 D C1
and 1. The supremum (resp. infimum) of the empty set is defined to be 1
(resp. 1), and the supremum (resp. infimum) of a set with no upper bound (resp. no
lower bound) is defined to be 1 (resp. 1). Accordingly, it is customary to write
1 < a < C1 for any real number a, and to define 1 C 1 D 1 .1/ D 1,
and .1/ C .1/ D .1/ 1 D 1, a ˙1 D ˙1 resp. a ˙1 D 1
for a > 0 resp. a < 0. It is important to keep in mind, however, that the symbols 1,
1 are not real numbers, and expressions such as 1 1 or 0 1 are undefined
(although see Section 6 of Chapter 4 for an exception).
If M is a subset of R and sup.M / 2 M (resp. inf.M / 2 M ), we say that the
supremum (resp. infimum) is attained, and speak of a maximum resp. minimum. In
this case, we may use the notation
max M; min M:
It is important to keep in mind that, unlike the supremum and infimum, a maximum
and/or minimum of a non-empty bounded subset of R may not exist. A non-empty
finite subset of R, however, always has a maximum and a minimum.
Variants of notation associated with suprema and infima (resp. maxima and
minima) are often used. For example, instead of sup M , one may write
sup x;
x2M
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6 1 Preliminaries
Let us fix notations for open and closed intervals in R: As usual, .a; b/ means the
set of all x 2 R such that a < x < b, where a; b are real numbers or ˙1. We will
denote by ha; bi the corresponding closed interval, i.e. the set of all x 2 R [ f˙1g
such that a x b. The reader can fill in the meaning of the symbols ha; b/, .a; bi.
1.2
order.
One introduces the complex conjugate of x D .x1 ; x2 / as x D .x1 ; x2 /. It is
easy to see that
Further, there is the absolute value (also called the modulus) defined by setting
p q
jxj D xx D x12 C x22
x
(thus, x 1 D ).
jxj2
If we view C as the Euclidean plane (one often speaks of the Gaussian plane)
then jxj is the standard distance of x from .0; 0/, and jx yj is the standard
Pythagorean distance.
Usually one sets i D .0; 1/ and writes
(note that the multiplication rule in C comes from distributivity and the equality
i 2 D 1). In the other direction, one puts
and calls these real numbers the real resp. imaginary part of x1 C ix2 .
We have a natural embedding of fields
.x 7! .x; 0// W R ! C
which will be used without further mention; note that this embedding respects the
absolute value.
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1 Real and complex numbers 7
1.3 Theorem. For the absolute value of complex numbers one has
jx C yj jxj C jyj:
Proof. Let x D x1 C ix2 and y D y1 C iy2 . We can assume y ¤ 0. For any real
number we have 0 .xj C yj /2 D xj2 C 2xj yj C 2 yj , j D 1; 2. Adding
these inequalities, we obtain
x1 y1 C x2 y2
Setting D yields
jyj2
1.3.2 Comment:
A function is basically the same thing as a map, although in many texts (including
this one), the term function is reserved for a map whose codomain is a set whose
elements we perceive as numbers, or at least some closely related generalizations.
For example, the codomain may be R, C or a subset of one of these sets, or it may
be, say, Œ0; 1. Sometimes, we will allow the codomain to consist even of n-tuples
of numbers, see for example Chapter 3. While many basic courses define functions
simply by formulas without worrying about the domain and codomain, in a rigorous
view of the subject, specifying domains and codomains is essential for capturing
even the most basic phenomena: Consider, for example, the function
f .x/ D x 2 : (*)
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8 1 Preliminaries
1.4.1
Recall that a polynomial with coefficients in R resp. C is an expression which is
either
.x 7! an x n C C a1 x C a0 / W R ! R resp. C ! C:
The zero polynomial determines a function, too, namely one which is constantly 0.
In analysis, it is quite common to identify a polynomial with the function it
determines (although note carefully that the domain and codomain of the function
corresponding to a polynomial with real coefficients will change if its coefficients
are considered as complex numbers). Nevertheless, this identification is permissible,
since two different polynomials over R (resp. C) never correspond to the same
function. To this end, note that it suffices to show that a non-zero polynomial does
not correspond to the 0 function (by passing to the difference). To this end, simply
note that if jx0 j is very large, then
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1 Real and complex numbers 9
Moreover, q.x/ has degree n 1. If the coefficients of p.x/ and the number c are
real, then the coefficients of the polynomial q.x/ are real.
Proof. For existence, recall (or observe by chain cancellation) that for k 2 N,
Therefore,
p.x/ p.c/ D an .x n c n / C C a1 .x c/
By Lemma 1.4.2, we then see that every polynomial of degree n with coefficients
in C can be written uniquely (up to order of factors) as
p.x/ D an .x c1 / .x cn /
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10 1 Preliminaries
2.1
We write
The reader is certainly familiar with the easy facts such as lim.xn C yn / D lim xn C
lim yn or lim.xn yn / D lim xn lim yn , etc.
2.2
This set is non-empty (a 2 M ) and bounded (no x > b is in M ) and hence there is
a finite s D sup M . By the definition, each
1 1
Kn D fk j s < xk < s C g
n n
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3 Continuous functions 11
is infinite, and we can choose, first, xk1 such that s 1 < xk1 < s C 1 and if
k1 < < kn are chosen with kj 2 Kj we can choose a knC1 2 KnC1 such that
knC1 > kn . Then obviously limn xkn D s, and equally obviously a s b. t
u
2.4 Theorem. (Bolzano - Cauchy) Every Cauchy sequence of real numbers con-
verges.
Proof. Since for some m and all n m, jxn xm j < 1, a Cauchy sequence is
bounded and hence it contains a subsequence xk1 ; : : : ; xkn ; : : : converging to an x.
But then limn xn D x: indeed, choose for an " > 0 an n0 such that for m; n n0
we have jxm xn j < " and jx xkn j < ". Then, since kn n, jx xn j < 2" for
n n0 . t
u
2.5
From 1.3.1, we see that if .xn D xn1 C ixn2 /n is a sequence of complex numbers
then
.xn /n converges if and only if both .xnj /n converge
and
.xn /n is Cauchy if and only if both .xnj /n are Cauchy.
Consequently we can infer from Theorem 2.4 the following
Corollary. Every Cauchy sequence of complex numbers converges.
3 Continuous functions
3.1
Recall that a real (resp. complex) function of one real (resp. complex) variable is a
mapping
In the real case X will be most often an interval, that is, a set J R such that
x; y 2 J and x z y implies that z 2 J .
Recall the standard notation from 1.1 for (bounded) open and closed intervals:
The intervals ha; bi will be often referred to as compact intervals; the reason for this
terminology will become apparent in Chapter 2 below. A function f W X ! R resp.
C is said to be continuous if
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12 1 Preliminaries
8x 2 X 8" > 0 9ı > 0 such that jy xj < ı ) jf .y/ f .x/j < ": (3.1.1)
Proof. for the maximum. Set M D ff .x/ j x 2 ha; big. If it is not bounded choose
xn > n and consider a convergent subsequence xkn with limit y. We have f .y/ D
limn xkn which is impossible because it would yield f .y/ > n for all n. Hence M
is bounded and has a finite supremum s. Now choose xn with s n1 < xn s, and
a convergent subsequence xkn with limit y 2 ha; bi to obtain f .y/ D s. t
u
3.5
8" > 0 9ı > 0 such that 8x; y; jy xj < ı ) jf .y/ f .x/j < ":
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4 Derivatives and the Mean Value Theorem 13
Proof. Suppose not. Then there exists an " > 0 such that
1
8n 9xn ; yn such that jxn yn j < and jf .xn / f .yn /j ":
n
Choose a convergent subsequence .xkn /n and then a convergent subsequence .ykmn /n
of .ykn /n . Then we have limn xkmn D limn ykmn contradicting Proposition 3.2 and
the inequality j limn f .xkmn / limn f .ykmn /j ". t
u
4.1
lim f .x/ D A
x!a
Note that f does not have to be defined in a, and if it is, lim f .x/ D A does not
x!a
say anything about the value f .a/.
4.2
f .x C h/ f .x/
lim DA
h!0 h
(that is, if the limit on the left-hand side exists, and if it is equal to a). The reader is
certainly familiar with the notation
df .x/
A D f 0 .x/; or
dx
and with the basic computation rules like .f C g/0 D f 0 C g 0 or .fg/0 D f 0 g C
fg 0 etc.
4.3 Theorem. A function f has a derivative A at the point x if and only if there is
a function defined on some .ı; ı/ X f0g (ı > 0) such that
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14 1 Preliminaries
f .x C h/ f .x/
D A C .h/
h
f .x C h/ f .x/
and hence lim D A. On the other hand, if the derivative exists
h!0 h
f .x C h/ f .x/
then we can set .h/ D A. t
u
h
4.4.1 Theorem. (Rolle) Let f be continuous in ha; bi and let it have a derivative
in .a; b/. Let f .a/ D f .b/. Then there is a c 2 .a; b/ such that f .c/ D 0.
Proof. If f is constant then f 0 .c/ D 0 for all c. If not then, as f .a/ D f .b/, either
its maximum or its minimum (recall Theorem 3.4) has to be attained in a c 2 .a; b/.
By 4.3.1, f 0 .c/ D 0. t
u
4.4.2 Theorem. (The Mean Value Theorem, Lagrange’s Theorem) Let f be contin-
uous in ha; bi and let it have a derivative at .a; b/. Then there is a c 2 .a; b/ such
that
f .b/ f .a/
f 0 .c/ D :
ba
More generally, if, furthermore, g is a function with the same properties and such
that g.b/ ¤ g.a/ and g 0 .x/ ¤ 0 then there is a c 2 .a; b/ such that
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4 Derivatives and the Mean Value Theorem 15
4.4.3
The Mean Value Theorem is often used in the following form (to be compared
with 4.3):
let x; x C h be both in an interval in which f has a derivative. Then
4.5.1
A function f is said to be convex resp. concave on an interval ha; bi if for any two
x < y in ha; bi and any z D tx C .1 t/y, (0 < t < 1), between these arguments,
(that is, the points of the graph of f lay below (resp.above) the straight line
connecting the points .x; f .x// and .y; f .y//).
Let the second derivative be non-negative. Then we have x < u < z < v < y and
u < w < v such that
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16 1 Preliminaries
so that
ap bq
ab C :
p q
Proof. Since ln00 .x/ D x12 < 0, ln is concave; thus if, say ap < b q we have
1 1 1 1
ln. ap C b q / ln.ap / C ln.b q / D ln a C ln b D ln.ab/
p q p q
Just as we defined the first and second derivative of a function on an open interval
J , we may iterate the process to define the third, fourth derivative, etc. In general,
we speak of the derivative of n’th order, and define
(Of course, as before, for a given function, such higher derivatives may or may not
exist.)
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4 Derivatives and the Mean Value Theorem 17
X
n
f .k/ .a/ f .nC1/ .c/
f .x/ D .x a/k C .x a/nC1 :
kŠ .n C 1/Š
kD0
Proof. Fix x and a and define a function R.t/ of one real variable t by setting
X
n
f .k/ .t/
R.t/ D f .x/ .x t/k :
kŠ
kD0
Then we have
X
n
f .kC1/ .t/ X
n1
f .lC1/ .t/ f .nC1/ .t/
R0 .t/ D .x t/k C .x t/l D .x t/n :
kŠ lŠ nŠ
kD0 lD0
Now define g.t/ D .x t/nC1 . Then g 0 .t/ D .n C 1/.x t/n and g.x/ D 0. Since
also R.x/ D 0 we obtain from Theorem 4.4.2,
and hence
X
n
f .k/ .a/
and the statement follows, since R.a/ D f .x/ .x a/k , that is,
kŠ
kD0
X
n
f .k/ .a/
f .x/ D .x a/k C R.a/. t
u
kŠ
kD0
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18 1 Preliminaries
Theorem. Let f be a function such that f 0 and f 00 exist and are continuous on
an open interval .a; b/ containing a point x0 . Suppose further that f 0 .x0 / D 0,
f 00 .x0 / < 0 (resp. f 00 .x0 / > 0). Then x0 is a local maximum (resp. local minimum)
of f .
Proof. Let us treat the case of f 00 .x0 / D q > 0; the proof in the other case is
analogous. By Taylor’s Theorem, for x 2 .a; b/, x ¤ x0 , there exists a point c in
the open interval between x0 and x such that
f 00 .c/
f .x/ D f .x0 / C .x x0 /2 : (*)
2
Since f 00 is continuous, there exists a ı > 0 such that for x 2 .x0 ı; x0 C ı/,
f 00 .c/ > 0. Then it follows immediately from (*) that if x 2 .x0 ı; x0 C ı/,
x ¤ x0 , f .x/ > f .x0 /. t
u
5 Uniform convergence
5.1
8" > 0 9n0 such that 8n n0 8x; jfn .x/ f .x/j < ":
Proof. Take an x0 2 X and an " > 0. Choose an n such that for all n n0 and
for all x, jfn .x/ f .x/j < 3" , and then a ı > 0 such that jfn .x0 / fn .x/j < 3" for
jx0 xj < ı. Then for jx0 xj < ı,
jf .x0 / f .x/j jf .x0 / fn .x0 /j C jfn .x0 / fn .x/j C jfn .x/ f .x/j < ":
t
u
5.3 Theorem. Let fn have derivatives on an open interval J , let fn ! f and let
fn0 g. Then f has a derivative and f 0 D g.
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6 Series. Series of functions 19
Proof. By the Mean Value Theorem we have for some 0 < < 1,
ˇ ˇ
ˇ f .x C h/ f .x/ ˇ
ˇ g.x/ ˇ
ˇ h ˇ
ˇ ˇ
ˇ f .x C h/ fn .x C h/ f .x/ fn .x/ fn .x C h/ fn .x/ ˇ
Dˇ ˇ C C g.x/ˇˇ
h h h
ˇ ˇ
ˇ f .x C h/ fn .x C h/ f .x/ fn .x/ ˇ
Dˇ ˇ C C fn .x C h/ g.x/ˇˇ
0
h h
1 1
jf .x C h/ fn .x C h/j C jf .x C h/ fn .x C h/j
jhj jhj
C jfn0 .x C h/ g.x C h/j C jg.x C h/ g.x/j:
Fix an h ¤ 0 such that jg.x C h/ g.x/j < 4" . Then choose an n such that
(1) jf .x C h/ fn .x C h/j < 4" jhj and jf .x/ fn .x/j < 4" jhj, and
(2) jfn0 .x C h/ g.x C h/j < 4" .
(Inequality (2) is where we need the convergence to be uniform: we do not know
the exact position of x C h). Then
ˇ ˇ
ˇ f .x C h/ f .x/ ˇ
ˇ g.x/ ˇ < 1 " jhj C 1 " jhj C " C " D ": t
u
ˇ h ˇ jhj 4 jhj 4 4 4
6.1
Let .an /n be a sequence of real or complex numbers. The associated series (or sum
P
1 P
of a series) an (briefly, an if there is no danger of confusion) is the limit
nD1
X
n
P
lim ak provided it exists; in such a case we say that an converges, and we say
n
kD1 P
that it converges absolutely if jan j converges.
6.2.1 Proposition.
P An absolutely convergent
P series converges. More generally, if
jan j bn and bn converges then an converges.
X
n X
n
Proof. Set sn D ak and s n D bk . For m n we have
kD1 kD1
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20 1 Preliminaries
X
m X
m X
m
jsm sn j D j an j jan j bn D js m s n j:
kDnC1 kDnC1 kDnC1
X
m
Proof. The formula is equivalent to stating that jak j < " for n0 n m.
kDn
X
n
Thus, the condition amounts to stating that the sequence . jak j/n is
kD1
Cauchy. t
u
P
6.2.3 Theorem. Let an converge absolutely. Then for all bijections p from the
1
X
set of natural numbers f1; 2; : : : g to itself the sums ap.n/ are equal.
nD1
1
X
Proof. Let ap.n/ D s for a bijection p. Choose n1 sufficiently large such that
X "
nD1
jan j < for every finite K fn j n n1 g and, further, an n0 such that for
2
k2K
n n0 we have
ˇ n ˇ
ˇX ˇ "
ˇ ˇ
ˇ ap.n/ s ˇ < and fp.1/; : : : ; p.n/g f1; : : : ; n1 g:
ˇ ˇ 2
kD1
t
u
6.3
It is worth taking this a little further. A set S is called countable if there exists a
bijection W f1; 2; : : : g ! S . Note that this is the same as ordering S into an
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6 Series. Series of functions 21
X
sup jas j
KS finite s2K
1
X
is finite. By Proposition 6.2.2, this is equivalent to a .n/ converging absolutely
nD1
for one specified bijection (which can be arbitrary). Theorem 6.2.3 then shows
that when this occurs, then
X
as
s2S
X and let S D
6.3.1 Theorem. Let S1 ; S2 ; : : : be disjoint finite or countable sets,
[
Si . Then the set S is finite or countable. Furthermore, if as converges
i s2S
absolutely, then
0 1
1
X X X
@ as A D as ; (*)
i D1 s2Si s2S
Proof. The case when S is finite is not interesting. Otherwise, we may order the
elements of S into an infinite sequence as follows: Assume each of the sets Si is
ordered into a (finite or infinite) sequence. Then let Tn consist of all the i ’th elements
S of Sj such that 1 i; j n. Then clearly each Tn is finite, and Tn TnC1 ,
(if any)
and Ti D S . Thus, we can order S by taking all the elements of T1 , then all the
remaining elements of T2 , etc. Thus, S is countable. X
Now let us investigate (*). The supremum sup jas j over finite subsets K
s2K
of Si is less than or equalX
to the analogous supremum over K finite subsets of
S , which shows that each as converges absolutely. Further, for a finite subset
s2Si
K 1; 2; : : : ,
ˇ ˇ
X ˇˇ X ˇˇ X X XX
ˇ as ˇˇ jas j sup jas j
ˇ
i 2K ˇs2Si ˇ i 2K s2Si i 2K s2Li
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22 1 Preliminaries
where the supremum on the right-hand side is over all finite subsets Li Si . We
see that the right-hand side is finite by our assumption of absolute convergence over
S , and therefore the left-hand side of (*) converges absolutely.
Finally, to prove equality in (*), use a variation of the above proof of the fact that
S is countable: Let Tn consist of sufficiently many elements of S1 ; : : : ; Sn such that
the sum
0 1
Xn X
@ as A
i D1 s2Tn \Si
differs from
0 1
X
n X
@ as A
i D1 s2Si
by less than 1=n. Then the limit of these particular partial sums is the left- hand side
of (*), but is also equal to the right-hand side by absolute convergence. t
u
1
X 1
X
6.3.2 Corollary. Let am , bn be absolutely convergent series. Then
mD0 nD0
1
! 1
! 1
!
X X X X
n
am bn D ak bnk ; (*)
mD0 nD0 nD0 kD0
6.4
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7 Power series 23
P X
n
all x 2 X we say that fn converges (resp. converges absolutely). If fk .x/
kD1
1
X
f .x/ we say that fn converges uniformly.
nD1
6.4.1
Since finite sums of continuous functions are continuous and since .f1 C Cfn /0 D
f10 C C fn0 we obtain from Theorem 5.2, Theorem 5.3 and Proposition 6.2.1
P
Corollary. 1. Let fn be continuous and let fn uniformly converge. Then the
resulting P
function is continuous. P
2. Let f D n fn converge, P let fn0 exist and let fn0 converge
P uniformly. Then f 0
0
exists and is equal to fn (that is, the derivative of fn can be obtained by
taking derivatives of the individual summands). P
3. The statements 1 (resp 2) apply to the case of jfn .x/j an with an
convergent; here the convergence is, moreover, absolute.
7 Power series
7.1
1
X
A power series with center c is a series an .x c/n . So far we will limit ourselves
nD1
to the real context; later in Chapter 10, we will discuss them in the complex case.
7.2
The limes superior (sometimes also called the upper limit) of a sequence .an /n of a
real number is the number
It obviously exists if the sequence .an /n is bounded; if not we set lim supn an D
C1. It is easy to see that lim supn an D lim an whenever the latter exists. The limes
inferior (or lower limit) is defined analogously with inf and sup switched.
7.2.1 Proposition. Let lim sup an D inf sup an D a and limn bn D b. Let an ; bn
n kn
0 and let a; b be finite. Then lim sup an bn D ab.
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24 1 Preliminaries
Proof. Choose an " > 0 and a K > a C b. Take an > 0 such that K > a C b C
and K < " There is an n0 such that
That is, for every n n0 there exists a k.n/ n such that a ak.n/ < a C and
b < bk.n/ < b C so that
and since ab " < ab K < ab a and .a C b C / < K < " we see that
7.2.2
1
X
For a power series an .x c/n define the radius of convergence
nD1
1
D ..an /n / D p
lim sup n
jan j
p
if lim sup n
jan j ¤ 0; otherwise set ..an /n / D C1.
1
X
Theorem. Let r < ..an /n /. Then the power series an .x c/n converges
n1
absolutely and uniformly on the set fx j jx cj rg.
1
X
On the other hand, if jx cj > then an .x c/n does not converge.
nD1
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7 Power series 25
7.3
1
X
Obviously it converges if and only if nan .x c/n does and hence its radius of
nD1
convergence is
1
p :
lim sup n njan j
p p p p
By Proposition 7.2.1, lim sup n njan j D lim sup n n n jan j D lim n n lim sup
p p p 1
n
jan j D lim sup n jan j (since lim n n D lim e n ln n D e 0 D 1). Thus,
from 5.3 and 6.3.1 that for jx cj < the series an .x c/ has a derivative,
and that it is obtained as the sum of the derivatives of the individual summands.
So far, this derivative had to be understood as in the real context. In fact, however,
it is valid for complex power series as well; see Chapter 10.
7.4 Remark
In particular
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26 1 Preliminaries
f .k/ .c/
f .k/ .c/ D nŠan ; and hence ak D :
kŠ
Thus, if a function can be written as a power series with a center c then the
coefficients an are uniquely determined (they do depend on the c, of course).
Compare this with the formula in 4.6.1. It should be noted, though, that in real
analysis it can easily happen that a function f has all derivatives without being
.nC1/
representable as a power series: the remainder f .nC1/Š.t / .x c/nC1 may not converge
to zero with increasing n (see Exercise (13)). In fact, it is interesting to note that
many important constructions in real analysis, such as the smooth partition of unity
which we will need in Chapter 12, depend on the use of such functions.
8.1
The mesh of the partition is the maximum of the numbers jti C1 ti j. A partition D 0 W
a D t00 < t10 < < tm0 D b refines D if ftj j j D 1; : : : ; ng ftj0 j j D 1; : : : ; mg.
Let f W ha; bi ! R be a bounded function (this means that the set of values of
f is bounded). Define the lower and upper sum of f in D as
X
n X
n
s.f; D/ D mj .tj tj 1 / and S.f; D/ D Mj .tj tj 1 /
j D1 j D1
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8 A few facts about the Riemann integral 27
8.2
By Proposition 8.1.1, we can define the lower resp. upper Riemann integral of f
over ha; bi by setting
Z b Z b
Rb Rb
If f .x/dx D a f .x/dx we denote the common value by
a
Z b Z b
f .x/dx or briefly f
a a
II. Let the statement hold. Choose an " > 0 and a D such that S.f; D/s.f; D/ >
". Then
Z b Z b
f S.f; D/ < s.f; D/ C " f C ":
a a
Rb Rb
Since " > 0 was arbitrary, af D f. t
u
a
8.3 Theorem. For every continuous function f W ha; bi ! R the Riemann integral
Rb
a f exists. In fact, more strongly, for every sequence Dn of partitions of ha; bi
whose mesh approaches 0 with n ! 1, we have
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28 1 Preliminaries
Z b
lim s.f; Dn / D lim S.f; Dn / D f:
n!1 n!1 a
Proof. Let " > 0. By 3.5.1, f is uniformly continuous. Hence there exist a ı > 0
such that
"
jx yj < ı ) jf .x/ f .y/j < :
ba
8.4 Theorem. (The Integral Mean Value Theorem) Let f be a continuous function
on ha; bi, M D maxff .x/ j x 2 ha; big and m D minff .x/ j x 2 ha; big (they
exist by 3.4). Then there exists a c 2 ha; bi such that
Z b
f .x/dx D f .c/.b a/:
a
Rb
Thus there is a K, m K M such that a f .x/dx D K.b a/. By 3.3, there
exists a c such that K D f .c/. t
u
8.5 Proposition. Let a < b < c and let f be a bounded function defined on ha; ci.
Then
Z b Z c Z c Z b Z c Z c
f C f D f and f C f D f:
a b a a b a
Proof. Denote by D.u; v/ the set of all paritions of hu; vi. For D1 2 D.a; b/
and D2 2 D.b; c/ define D1 C D2 2 D.a; c/ as a union of the two sequences.
Obviously
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8 A few facts about the Riemann integral 29
We have
Z b Z c
f C f D sup s.D1 ; f / C sup s.D2 ; f /
a b D1 2D.a;b/ D2 2D.b;c/
8.5.1 Convention Rb Ra
For b < a we will write formally a f for b f . Then we have, for any a; b; c,
Z b Z c Z c
f C f D f:
a b a
1
.F .x C h/ F .x// D f .x C h/
h
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30 1 Preliminaries
Rx Rb Rb Ra
(By 4.4.4, a f .t/dt G.x/ is constant. Thus, a f D a f a f D G.b/ C
C .G.a/ C C / D G.b/ G.a/.)
9 Exercises
(1) Assuming the Fundamental Theorem of Algebra, prove that every non-zero
polynomial with coefficients in R is a product of polynomials with coefficients
in R each of which has degree 2. [Hint: Use 1.4.4.]
(2) Prove that the set R of all real numbers is not countable (we say it is
1
X
uncountable). [Hint: Prove that the numbers ak 2k are all well-defined
kD0
1
X
and different for all choices ak 2 f0; 1g. If there were a sequence ak;n 2k ,
kD0
1
X
n 2 N of all these numbers, then the number .1ak;k /2k would be different
kD0
from all of them - a contradiction.]
X 1
(3) (a) Prove directly that the function e x D x n satisfies e x e y D e xCy .
nŠ
[Hint: Use Corollary 6.3.2]
(b) Prove that e x ¤ 0 for any x 2 R. [Hint: use (a).]
(c) Prove that e x is a continuous function on R which takes on only positive
values. [Hint: Use Theorem 7.2.2, Theorem 5.2 and Theorem 3.3.]
(4) Using the definition from Exercise (3), prove that .e x /0 D e x . [Hint: Corollary
6.4.1 is relevant.]
(5) (a) Prove that e x is an increasing function on R. [Hint: Use Exercises (3)
and (4).]
(b) Prove that lim e x D 0, lim e x D 1. [Hint: Use (a) and Exercise (3).]
x!1 x!1
(6) (a) Prove that there exists a function ln.x/ W fx 2 Rjx > 0g ! R inverse to
e x . [Hint: Use Exercise (5) (b).]
(b) Prove that .ln.x//0 D 1=x. [Hint: This follows from the chain rule; a direct
proof can also be given using Theorem 4.3.]
(7) For a 2 R, x > 0, define x a D e a ln.x/ . Using the chain rule, prove that
.x a /0 D ax a1 .
X1
(8) Define functions sin.x/, cos.x/ by sin.x/ D .1/n x 2nC1 =.2n C 1/Š,
nD0
1
X
cos.x/ D .1/n x 2n =.2n/Š.
nD0
(a) Prove that cos.x/ D cos.x/; sin.x/ D sin.x/ (i.e. cos.x/ is even and
sin.x/ is odd).
(b) Prove that .sin.x//0 D cos.x/, .cos.x//0 D sin.x/. [Hint: Corollary
6.4.1 is relevant.]
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9 Exercises 31
(9) Prove that there exists a minimum number a > 0 such that cos.a/ D 0.
This number a is called =2. Prove that cos.x/ is decreasing in the interval
.0; =2/. [Hint: By Exercise (8), we have cos00 .x/ D cos.x/, while
cos.0/ D 1, .cos.0//0 D 0. This means that .cos.x//0 is negative in some
interval .0; "/, " > 0, and .cos.x//0 is decreasing on any interval .0; a/ on
which cos.x/ > 0. Let cos0 ."=2/ D b, cos."=2/ D c, b; c > 0. Then
cos."=2 C t/ c bt if "=2 < "=2 C t < a (a as above). From this, it follows
we cannot have a "=2 > c=b.]
(10) (a) Prove that cos.x ˙ y/ D cos.x/ cos.y/ sin.x/ sin.y/, sin.x ˙ y/ D
sin.x/ cos.y/ ˙ cos.x/ sin.y/. [Hint: analogous to Exercise (3).]
(b) Prove that sin. =2/ D 1, sin.x/ is increasing on the interval .0; =2/, and
cos. =2 x/ D sin.x/. [Hint: Let sin. =2/ D a. Apply (a) to show that
cos. =2 x/ D a sin.x/, sin. =2 x/ D a cos.x/, and therefore a2 D 1.
Observe that we must then have a D 1 because sin.x/ is increasing on the
interval .0; =2/ by Exercise (8).]
(11) Prove that cos.x/ and sin.x/ are both periodic with period 2 , their values
(on x real) are between 1 and 1, and describe their maxima and minima, and
intervals on which they are decreasing resp. increasing.
[Hint: Use Exercise (10) and the fact that cos.x/ is even to prove that cos.x C
/ D cos.x/, etc.]
(12) Now consider the definition of e x from Exercise (3) for a complex number x.
(a) Prove that e x is well-defined (i.e. the series converges) for all x 2 C, and
that e x e y D e xCy for x; y 2 C. [Interpret this as separate statements
about the real and imaginary parts.]
(b) Prove that for a complex number , the functions Re.e x /, Im.e x / are
continuous and differentiable in the real variable x, and that .e x /0 D e x
[this is, again, to be interpreted as equalities of the real and imaginary
parts].
e ix C e ix e ix e ix
(c) Prove the equalities cos.x/ D , sin.x/ D for x 2 R.
2 2i
[Remark: The attentive reader surely noticed that something is missing here;
we should learn how to differentiate with respect to a complex variable x.
(!) However, we will have to build up a lot more foundations, and wait until
Chapter 10 below, to understand that rigorously.]
(13) Let f .x/ D e 1=x for x > 0, f .x/ D 0 for x 0. Prove that f .n/ .0/ D 0 for
all n 1.
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1 Basics
1.1
Let RC denote the set of all non-negative real numbers and C1. A metric space
is a set X endowed with a metric (or distance function, briefly distance) d W X
X ! RC such that
(M1) d.x; y/ D 0 if and only if x D y,
(M2) d.x; y/ D d.y; x/, and
(M3) d.x; y/ C d.y; z/ d.x; z/.
Condition (M3) is called the triangle inequality; the reader will easily guess why.
The elements of a metric space are usually referred to as points.
Very often one considers distance functions which take on finite values only, but
allowing infinite distances comes in handy sometimes.
1.1.1 Examples
(a) The set R of real numbers with the distance function d.x; y/ D jx yj.
(b) The set (plane) C of complex numbers, again with the distance jx yj; note,
however, that here the fact that it satisfies the triangle inequality is much less
trivial than in the previous case (see Theorem 1.3 of Chapter 1).
(c) The Euclidean space Rm D f.x1 ; : : : ; xm / j xj 2 Rg
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34 2 Metric and Topological Spaces I
r
X
d..x1 ; : : : ; xm /; .y1 ; : : : ; ym // D .xj yj /2 :
Comment: In linear algebra, there are good reasons for distinguishing row and
column vectors, and equally good reasons why the ordinary Eucliean space
Rn should consist of column vectors. This is the reason why we used the
subscript Rn above for row vectors, which are easier to write down (compare
with A.7.3). From the point of view of metric and topological spaces, however,
the distinction between row and column vectors has no meaning. Because of
that, in this chapter, we will use the symbols Rn and Rn interchangably, not
distinguishing between row and column vectors.
(d) C.ha; bi/, the set of all continuous real functions on the interval ha; bi, with
S 1 D f.x; y/ 2 R2 j x 2 C y 2 D 1g
where for two points P; Q 2 S 1 , d.P; Q/ is the lesser of the two angles
between the lines ftPjt 2 Rg and ftQjt 2 Rg.
(g) Any set S with the metric given by d.x; y/ D 0 if x D y 2 S and d.x; y/ D 1
if x ¤ y 2 S . This is known as the discrete space.
1.2 Norms
The metrics in Examples 1.1.1 (a)–(e) in fact all come from a more special situation,
which plays an especially important role. A norm on a vector space V (over real or
complex numbers) is a mapping jj jj W V ! R such that
(1) jjxjj 0, and jjxjj D 0 only if x D o,
(2) jjx C yjj jjxjj C jjyjj, and
(3) jj˛xjj D j˛j jjxjj.
1.2.1
A normed vector space is a (real or complex) vector space V provided with a norm.
(The term normed linear space is also common.) Since we have
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1 Basics 35
the function .x; y/ D jjx yjj is a metric on V , called the metric associated with
the norm. In this sense, we can always view a normed linear space as a metric space.
1.2.2 Examples
1. Any of the following formulas yields a norm in Rn .
(a) jjxjj D P
max xj ,
(b) jjxjj D q jxj j,
P 2
(c) jjxjj D xj .
Notice that (c) gives the metric space in Example 1.1.1 (c).
2. In the space of bounded real functions on a set X we can consider the norm
jj'jj D supfj'.x/j j x 2 X g:
Indeed: (1) of 1.2 is obvious. Further, by the Cauchy-Schwarz inequality (see 4.4 of
Appendix A),
1.3 Convergence
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36 2 Metric and Topological Spaces I
1.3.1 Examples
(a) The usual convergence in R or C.
(b) Consider the examples in 1.1.1 (d) and (e). Realize that the convergence of a
sequence of functions f1 ; f2 ; : : : in these spaces is what one usually calls uniform
convergence of functions.
1.4
Two metrics d1 ; d2 on the same set X are said to be equivalent if there exist positive
real numbers ˛; ˇ such that for every x; y 2 X ,
1.5
for every x 2 X and every " > 0 there is a ı > 0 such that, for
every y in X ,
(ct)
d.x; y/ < ı ) d 0 .f .x/; f .y// < ":
for every " > 0 there is a ı > 0 such that, for all x; y in X
(uct)
d.x; y/ < ı ) d 0 .f .x/; f .y// < ":
Note the subtle difference between the two concepts. In the former the ı can depend
on x, while in the latter it depends on the " only. For example,
f D .x 7! x 2 / W R ! R
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2 Subspaces and products 37
It is easy to prove
1.5.2
Here is another easy but important
1.6 Proposition. A map f W .X; d / ! .Y; d 0 / is continuous if and only if for every
convergent sequence .xn /n in .X; d /, the sequence .f .xn //n is convergent and
Proof. ): Let lim xn D x. Consider the ı > 0 from (ct) taken for the x and an
" > 0. There is an n0 such that n n0 implies d.xn ; x/ < ı. Then for n n0 ,
d 0 .f .xn //; f .x// < ".
(: Suppose f is not continuous. Then there is an x 2 X and an "0 > 0
such that for every ı > 0 there exists an x.ı/ such that d.x.ı/; x/ < ı while
d 0 .f .x.ı// "0 . Now set xn D x. n1 /; obviously lim xn D x and .f .xn //n does not
converge to f .x/. t
u
2.1
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38 2 Metric and Topological Spaces I
Proof. 1. For " > 0 take ı D ". For the consequence recall 1.5.1 and the fact that
f jX 0 D fj .
2. For x and " > 0 use the same ı as for f . t
u
2.2
Y
m
Let .Xi ; di /, i D 1; : : : ; m, be metric spaces. On the cartesian product Xi D
i D1
X1
Xm consider the following distances:
v
u m
uX
..x1 ; : : : ; xm /; .y1 ; : : : ; ym // D t di .xi ; yi /2 ;
i D1
X
m
..x1 ; : : : ; xm /; .y1 ; : : : ; ym // D di .xi ; yi /; and
i D1
Obviously d..xi /i ; .yi /i / ..xi /i ; .yi /i /; ..xi /i ; .yi /i / and finally ..xi /i ;
X
m
.yi /i / maxj dj .xj ; yj / D n d..xi /i ; .yi /i /.
i D1
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3 Some topological concepts 39
2.2.2 Q
The space Xi endowed with any of the metrics , , d (typically, by d ) will be
referred to as the product of the spaces .Xi ; di /, i D 1; : : : ; m.
Y
Theorem. 1. The projections pj D ..X1 ; : : : ; xm / 7! xj / W .Xi ; di / !
i
.Xj ; dj / are uniformly continuous.
2. A sequence
.x11 : : : ; xm
1
/; .x12 : : : ; xm
2
/; .x13 : : : ; xm
3
/; : : : (*)
Q
converges in .Xi ; di / if and only if each of the sequences
3.1 Neighborhoods
.x; "/ D fy j d.x; y/ < "g:
.x; "/ U:
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40 2 Metric and Topological Spaces I
Remark: While the concept of an "-ball depends on the concrete metric, the
concept of neighborhood does not change if we replace a metric by an equivalent
one. In fact, we can change the metric even much more radically – see Exercise (5)
below.
3.2.1 Proposition. 1. X and ; are open. If U and SV are open then U \ V is open,
and if Ui , i 2 J , are open (J arbitrary) then Ui is open.
i 2J
2. U is open if and only if X X U is closed.
[ If A and B are closed then A[B is closed, and if Ai , i 2 J ,
3. X and ; are closed.
are closed then Ai is closed.
i 2J
3.3 Closure
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3 Some topological concepts 41
A D fx j d.x; A/ D 0g:
This definition seems to depend heavily on the distance function. But we have
3.3.1 Proposition. 1. The set A is closed, and it is the smallest closed set
containing A. In other words,
\
AD fB closed j A Bg:
Proof. 1 is trivial.
2: By 1, A [ B A [ B. Now let x 2 A [ B; x is or is not in A. In the latter
case, all sufficiently close elements from A [ B have to be in B and hence x 2 B.
3: By 3.3.1 1, A is closed and since it contains B D A, it also contains B D A.
t
u
3.4
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42 2 Metric and Topological Spaces I
f ŒA f ŒA:
f 1 ŒB f 1 ŒB:
3.5
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4 First remarks on topology 43
Very often, a choice of metric is not really important. We may be interested just
in continuity, and a concrete choice of metric may be somehow off the point. For
example, note that the ”natural” Pythagorean metric would have been a real burden
in dealing with the product. Sometimes it even happens that one has a natural notion
of continuity, or convergence, without having a metric defined first. It may even
happen that there is no reasonable way to define a metric.
This leads to a more general notion of a space, called a topological space. The
idea is to describe the structure of interest simply in distinguishing whether a subset
U X containing x “surrounds” (is a neighborhood of) x, or declaring some
subsets open resp. closed, or specifying an operator of closure. We will present here
three variants of the definition, which turn out to be equivalent.
4.1
We will start with the neighborhood approach, which was historically the first one
(introduced by Hausdorff in 1914). It is convenient to denote by P.X / the power
set of X , which means the set of all subsets of X (including the empty set and X ).
With every x 2 X , one associates a set U.x/ P.X /, called the system of the
neighborhoods of x, satisfying the following axioms:
(1) For each U 2 U.x/, x 2 U ,
(2) If U 2 U.x/ and U V X then V 2 U.x/,
(3) If U; V 2 U.x/ then U \ V 2 U.x/, and
(4) For every U 2 U.x/ and every y 2 V there is a V 2 U.x/ such that U 2 U.y/.
One then defines a (possibly empty) subset U of X to be open if U is a neighborhood
of each of its points. One defines a subset A of X to be closed if the complement
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44 2 Metric and Topological Spaces I
4.2
4.3
Or, one can start with a closure operator u W P.X / ! P.X / satisfying
(1) u.;/ D ; and A u.A/,
(2) u.A [ B/ D u.A/ [ u.B/ and
(3) u.u.A// D u.A/.
A is declared closed if u.A/ D A, the open sets are complements of the closed ones,
and U is a neighborhood of x if x … u.X X U /.
4.4
In fact one usually thinks of a topological space as a set endowed with all the
above mentioned notions simultaneously, and the only question is which of them
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4 First remarks on topology 45
one considers primitive concepts and which are defined afterwards. The resulting
structure is the same. (See the Exercises.)
4.5
4.6
f ŒA f ŒA:
f 1 ŒB f 1 ŒB:
Proof. Most of the implications can be proved by the same reasoning as in 3.4. The
only one needing a simple adjustment is
(5))(1): Let (5) hold and let V be a neighborhood of f .x/. Thus, f .x/ …
Y X V , that is, x … f 1 ŒY X V . Hence, U D X X f 1 ŒY X V D f 1 ŒV is
a neighborhood of x, and f ŒU D ff 1 ŒV V . t
u
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46 2 Metric and Topological Spaces I
4.7
B1 ; B2 2 B ) B1 \ B2 2 B and
[
for every U 2 ; U D fB j B 2 B; B U g:
(For example, the set of all open intervals, or the set of all open intervals with
rational endpoints are bases of the standard topology of the real line R).
One may wish to define a topological space where some particular subsets
are open, thus specifying a subset S P.X / of such sets without any a priori
properties. One easily sees that the smallest topology containing S is the set of all
unions of finite intersections of elements of S. Then one speaks of S as of a subbasis
of the topology obtained.
The preimages of (finite) intersections are (finite) intersections, and preimages of
unions are unions of preimages. Consequently we obtain from 4.6 an important
4.8
Let .X; / be a topological space and let Y X be a subset. We define the subspace
of .X; / carried (or induced) by Y as
.Y; jY / where jY D fU \ Y j U 2 g:
4.8.1 Convention
Unless otherwise stated, the subsets of a topological space will be understood to
be endowed with the induced topology, and we will subject the terminology to this
convention. Thus we will speak of “connected subsets” or “compact subsets” etc
(see below) or on the other hand of an ‘open subspace” or ”closed subspace”, etc.
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5 Connected spaces 47
5 Connected spaces
5.1
is connected.
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48 2 Metric and Topological Spaces I
The fact that the set R of all real numbers is connected is “intuitively obvious”, but
must be proved with care. Let us start with a preliminary result.
5.2.1 Lemma. Every open set U R is a union of countably (or finitely) many
disjoint open intervals.
where C are equivalence classes with respect to
are disjoint open intervals whose
union is U . t
u
5.2.2 Theorem. The connected subsets of R are precisely (open, closed, half-open,
bounded, unbounded, etc.) intervals.
Proof. Let us first prove that intervals are connected. Let J be an interval. Suppose
U; V are open in R, U \ V J , U \ V \ J D ;. Suppose U is non-empty. By
Lemma 5.2.1, U is a disjoint union of countably many open intervals Ui , i 2 I ¤ ;.
Without loss of generality, none of the sets Ui is disjoint with J . Choose i 2 I , and
suppose Ui D .a; b/ does not contain J . Then .a; b/ [ J is an interval containing
but not equal to .a; b/, so a 2 J or b 2 J . Let, without loss of generality, b 2 J .
Then b … V , b … Uj , j ¤ i , since V , Uj , j ¤ i are open and disjoint with Ui .
Thus, b 2 J X .U [ V /, which is a contradiction.
On the other hand, suppose that S R is connected but isn’t an interval. Then
there exist points x < z < y, x; y 2 S , z … S . But then S .1; z/ [ .z; 1/,
which contradicts the assumption that S is connected. t
u
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5 Connected spaces 49
Connected components may not be open: consider Q (with the topology induced
from R). Then the connected components are single points. We have, however,
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50 2 Metric and Topological Spaces I
The proof of the following result will seem, in nature, related to the proof of the
fact that the real numbers are connected. While this is true, it turns out to be mainly
due to special properties of the real numbers. The result itself is a reformulation of
compactness, a notion which we will discuss in the next section. An understanding
of this connection for general metric spaces, however, will have to be postponed
until Chapter 9 below.
By an open interval (resp. bounded closed interval) in Rn we mean a set of the
Y
n Y
n
form .ak ; bk / (resp. of the form hak ; bk i, 1 < ak ; bk < 1).
kD1 kD1
Proof. Let us first consider the case n D 1. Let ha; bi be contained in a union of a
set S open intervals. Let t 2 ha; bi be the supremum of the set M of all s 2 ha; bi
such that ha; si is contained in a union of some finite subset of S . We want to prove
that t D b. Assume, then, that t < b. Then there exists a J 2 S such that t 2 J . On
the other hand, by the definition of supremum, there exist si 2 M such that si % t.
Then, for some i , si 2 J . But we also know that there exists a finite subset F S
whose union contains ha; si i. Then the union of the finite subset F [ fJ g contains
ha; xi for every x 2 J , contradicting t D sup M .
Now let us consider general n. Assume, by induction, that the statement holds
with n replaced by n 1. Let K D ha1 ; b1 i
han ; bn i. Then for every point
x 2 ha1 ; b1 i, there exists, by the induction hypothesis, a finite subset Fx S
such that fxg
ha2 ; b2 i
han ; bn i Fx . Let Ix be the intersection of all the
(1-dimensional) intervals I1 where I1
In 2 Fx . Then ha1 ; b1 i is contained in
the union of the open intervals Ix , x 2 ha1 ; b1 i, and hence there are finitely many
[
k
points x1 ; : : : ; xk 2 ha1 ; b1 i such that ha1 ; b1 i Ixi . Then K is contained in the
i D1
union of the open intervals in Fx1 [ [ Fxk . t
u
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6 Compact metric spaces 51
(Apply the theorem to the set S of all open intervals which are contained in one
of the open sets in Q.)
6.1
6.2.1
Note that from the second part of the proof of the first statement we obtain an
immediate
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52 2 Metric and Topological Spaces I
preimages of closed sets being closed, also the images of closed sets are closed.
We will learn more about this phenomenon in Chapter 9 below. For now, let us
record the following
6.3 Proposition. Let X be a compact metric space. Then for each continuous real
function f on X there exist x1 ; x2 2 X such that
be a sequence of points of X
Y . In X , choose a convergent subsequence .xkn /n
of .xn /n . Now take the sequence .ykn /n in Y and choose a convergent subsequence
.ykrn /n . Then by 2.2.2.2 (and (1.2.1)),
A metric space .X; d / is bounded if there exists a number K such that for all
x; y 2 X , d.x; y/ < K. From the triangle inequality we immediately see that this
is equivalent to any of the following statements:
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6 Compact metric spaces 53
Proof. I. From Theorem 2.3 of Chapter 1, we already know that a bounded closed
interval is compact.
II. Now let X be a bounded closed subspace of Rm . Since it is bounded there are
intervals hai ; bi i, i D 1; ; : : : ; m, such that
X J D ha1 ; b1 i
ham ; bm i:
6.6
We have already observed that uniform continuity is a much stronger property than
continuity (even the real function x 7! x 2 is not uniformly continuous). But the
situation is different for compact spaces. We have
Proof. Let f be continuous but not uniformly continuous. Negating the defini-
tion,
there is an "0 > 0 such that for every ı > 0 there are x.ı/; y.ı/ such that
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54 2 Metric and Topological Spaces I
7 Completeness
7.1
Proof. 1. Let lim xn D x. For " > 0 choose an n0 such that d.xn ; x/ < "
2 for all
n n0 . Then for m; n n0 ,
" "
d.xm ; xn / d.xm ; x/ C d.x; xn / < C D ":
2 2
7.3
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7 Completeness 55
Proof. Let .xn /n be a Cauchy sequence in a compact metric space X . Then it has a
convergent subsequence, and by 6.2 2, it converges. t
u
7.5 Theorem. The Euclidean space Rm (in particular, the real line R) is complete.
Consequently, a subspace of Rm is complete if and only if it is closed.
fxn j n D 1; 2; : : : g J D ha1 ; b1 i
ham ; bm i
for sufficiently large intervals haj ; bj i. By 6.4 .xn /n converges in J and hence it
converges in Rm . u
t
Remark. The special case of the real line is the well-known Bolzano-Cauchy
Theorem (Theorem 2.4 of Chapter 1).
7.6
The following is the well-known Banach Fixed Point Theorem. At first sight it
may seem that its use will be rather limited: the assumption is very strong. But the
reader will be perhaps surprised by the generality of one of the applications in 3.3
of Chapter 6.
xnC1 D f .xn /:
C
d.xn ; xm / D C.q n1 Cq n C Cq m2 / C q n1 .1Cq Cq 2 C / D q n1 :
1q
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56 2 Metric and Topological Spaces I
Hence, .xn /n ia a Cauchy sequence and we have a limit x D lim xn . Now a mapping
f satisfying (*) is clearly continuous and hence we have
C.X /
the space of all bounded continuous real functions f W X ! R, endowed with the
metric
(We have d.f; g/ < " if and only if for all x 2 X , jf .x/ g.x/j < ".)
7.7.2 Proposition. The space C.X / with the metric defined above is complete.
Proof. Let .fn /n be a Cauchy sequence in C.X /. Then, since jfn .x/ fm .x/j
d.fn ; fm / for each x 2 X , every .fn .x//n is a Cauchy sequence in R, and hence a
convergent one. Set
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8 Uniform convergence of sequences of functions. Application: Tietze’s Theorems 57
and hence lim jfn .x/ fm .x/j D jfn .x/ lim fm .x/j D
m!1 m!1
jfn .x/ f .x/j 2" < ". Thus, for n n0 and for all x 2 X , jfn .x/
f .x/j < ". t
u
Proof of the Proposition continued. By the Claim and 8.2, f is continuous. Now
there exists an n0 such that for all n; m n0 , d.fn ; fm / D sup jfn .x/ fm .x/j <
x
1 and hence, taking the limit, we obtain jfn .x/ f .x/j 1 for all x. Thus, if
jfn0 .x/j K we have jf .x/j K for all x.
Now we know that f is bounded and continuous, hence f 2 C.X /, and by 7.7.1
and the Claim again, .fn /n converges to f in C.X /. t
u
7.7.3
Let a; b 2 R [ f1: C 1g. Put
On various occasions we have seen that general facts the reader knew about real
functions of one real variable held generally, and the proofs did not really need
anything but replacing jx yj by the distance d.x; y/. For example, this was
the case when studying the relationship between continuity with convergence, or
when proving that continuous maps of compact spaces are automatically uniformly
continuous; or the fact about maxima and minima of real functions on a compact
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58 2 Metric and Topological Spaces I
space (where in fact the general proof was in a way simpler, or more transparent,
due to the observation that the image of a compact space is compact).
In this section we will introduce yet another case of such a mechanical exten-
sion, namely the behavior of uniformly convergent sequences of mappings, resp.
uniformly convergent series of real functions. As an application we will present
rather important Tietze Theorems on extension of continuous maps.
8.1
f1 ; f2 ; f3 ; : : : W X ! Y
fn f:
8.1.1 Remarks
1. Note that if fn f then
The statement (*) alone, (called pointwise convergence), is much weaker, and
would not suffice as an assumption in 8.2 below.
2. Also note that in the above definition, one uses the metric structure in .Y; d 0 /
only. See 8.2.1 below.
8.2 Proposition. Let fn f for mappings .X; d / ! .Y; d 0 /. Let all the functions
fn be continuous. Then f is continuous.
Proof. For " > 0 choose n such that d 0 .fn .x/; f .x// < 3" for all x. Since fn is
continuous there is a ı > 0 such that d.x; y/ < ı implies d 0 .fn .x/f .x// < 3" . Now
we have the implication
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8 Uniform convergence of sequences of functions. Application: Tietze’s Theorems 59
8.2.1
Note that an analogous proposition also holds for a topological space .X; / instead
of a metric one. In the proof replace the requirement of ı by a neighborhood U of x
such that fn ŒU
.fn .x/; 3" / and use for y 2 U the triangle inequality as before.
P
8.3 Corollary. Let fn W .X; d / ! R be continuous functions, let an be a
convergent series of real numbers, and let for every n and every x, jfn .x/j an .
X n X1
Then gn .x/ D fk .x/ uniformly converge to fk .x/ and hence g D .x 7!
kD1 kD1
1
X
fk .x// is a continuous function.
kD1
8.4 Lemma. Let A; B be disjoint closed subsets of a metric space .X; d / and let
˛; ˇ be real numbers. Then there is a continuous function
' D ˆ.A; BI ˛; ˇ/ W X ! R
such that
'ŒA f˛g; 'ŒB fˇg and minf˛; ˇg '.x/ maxf˛; ˇg: (ˆ)
Proof. Set
d.x; A/
'.x/ D ˛ C .ˇ ˛/ :
d.x; A/ C d.x; B/
8.5 Theorem. (Tietze) Let A be a closed subspace of a metric space X and let
J be a compact interval in R. Then each continuous mapping f W A ! J can be
extended to a continuous g W X ! J (that is, there is a continuous g such that
gjA D f ).
Proof. For a degenerate interval ha; ai the statement is trivial and all the other
compact intervals are homeomorphic; if the statement holds for J1 and if h W J !
J1 is a homeomorphism we can extend for f W A ! J the hf to a g W X ! J1
and then take g D h1 g. Thus we can choose the J arbitrarily. For our purposes,
J D h1; 1i will be particularly convenient.
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60 2 Metric and Topological Spaces I
1 1
'1 D ˆ.A1 ; B1 I ; /:
3 3
We obviously have
2
8x 2 A; jf .x/ '1 .x/j :
3
Set f1 D f '1 .
Suppose we already have continuous
1 2
j'k .x/j ; fk .x/ D fk1 .x/ 'k .x/ and jfk .x/j : (*)
3k 3k
Then set
1 1 1 1
AnC1 D f 1 Œh ; i; BnC1 D f 1 Œh nC1 ; n i;
3n 3nC1 3 3
1 1
'nC1 D ˆ.AnC1 ; BnC1 I nC1 ; nC1 / and fnC1 D fn 'nC1 :
3 3
g W X ! h1; 1i:
f .x/ D '1 .x/Cf1 .x/ D '1 .x/C'2 .x/Cf2 .x/ D D '1 .x/C C'n .x/Cfn .x/
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8 Uniform convergence of sequences of functions. Application: Tietze’s Theorems 61
Proof. We can replace R by any space homeomorphic with R (recall the first
paragraph of the previous proof). We will take the open interval .1; 1/ instead
and extend a map f W A ! .1; 1/.
By 8.5, f can be extended to a g W X ! h1; 1i. Such g can, however reach the
values 1 or 1 and hence is not an extension as desired. To remedy the situation,
consider B D g 1 Œf1; 1g which is a closed set disjoint with A, consider the ' D
ˆ.A; B; 0; 1/ from 8.4, and define
Now we have f .x/ D g.x/ D g.x/ for x 2 A, and jg.x/j < 1 for all x 2 X : if
g.x/ D 1 or 1 then '.x/ D 0.
8.5.2
A subspace R of a space Y is said to be a retract of Y if there exists a continuous
r W Y ! R such that r.x/ D x for all x 2 R.
A metric space Y is injective if for every metric space X and closed A X ,
each continuous f W A ! Y can be extended to a continuous g W X ! Y . (Thus,
we have learned above that R and any compact interval are injective spaces.)
Proof. First we will prove that a Euclidean space itself is injective. Consider it as
the product
Rm D R
R m times
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62 2 Metric and Topological Spaces I
9 Exercises
d 0 .x; y/ D jx 3 y 3 j
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9 Exercises 63
(16) Path-connected components are defined the same way as connected com-
ponents in 5.4, with the word “connected” replaced by the word “path-
connected”. Are path-connected components necessarily closed? Prove or give
a counterexample.
(17) Check that convergence in the metric spaces defined in 1.1.1 (d), (e) is
precisely uniform convergence.
(18) Prove an analogue of Proposition 8.2 for uniform continuity instead of
continuity.
1
X
(19) Let K be the set of all real numbers of the form ak 3k , where ak 2 f0; 2g.
kD1
(This is called the Cantor set.) Prove that K is compact. Prove that K contains
no compact interval with more than one point.
(20) Prove that a subspace of Rm is injective if and only if it is a retract.
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In this chapter, we will learn multivariable differential calculus. We will develop the
multivariable versions of the concept of a derivative, and prove the Implicit Function
Theorem. We will also learn how to use derivatives to find extremes of multivariable
functions.
To understand Multivariable Differential Calculus, one must be familiar with
Linear Algebra. We assume that the typical reader of this book will already have
had a course in linear algebra, but for convenience we review the basic concepts in
Appendices A and B. We refer periodically to results of these Appendices, and we
recommend that the reader who has seen some linear algebra simply start reading the
present chapter, and refer to these results in the Appendix as needed. Notationally,
the most important are the conventions in Sections 1.3 and 7.3 of Appendix A below:
Rn will be the space of real n-dimensional column vectors (matrices of type n
1).
To avoid awkward notation, however, we will usually write rows and decorate them
with the superscript ‹T which means transposition (Subsection 7.3 in Appendix A.
Row or column vectors will be denoted by bold-faced letters, such as v. The zero
vector (origin) will be denoted by o.
1.1
We will deal with real functions of several real variables, that is, mappings
f W D ! R with a domain D Rn . Typically, D will be open. Intercheangably
f .x/ where, in accordance with convention 7.3 of Appendix A, x D .x1 ; : : : ; xn /T ,
we will also write f .x1 ; : : : ; xn /. When x 2 Rm , y 2 Rn , notations such as f .x; y/,
f .x; y1 ; : : : ; yn / will also be allowed for a function f of m C n variables.
Given such a function f , we will often be concerned with the associated
functions of one variable
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66 3 Multivariable Differential Calculus
Then each f .a; / and each f .; b/ is continuous, but f is not: the sequence . n1 ; n1 /
converges to .0; 0/ while lim f . n1 ; n1 / D 0 ¤ f .0; 0/.
1.2
Recall again Convention 1.3, 7.3 of Appendix A. It is important to note that a vector
function
f D .f1 ; : : : ; fm /T W D ! Rm ; fj W D ! R:
is continuous if and only if all the fi are continuous (recall Theorem 2.2.2 of
Chapter 2).
1.3 Composition
2.1
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2 Partial derivatives. Defining the existence of a total differential 67
@f .x1 ; : : : ; xn / @f
or .x1 ; : : : ; xn /;
@xk @xk
in case of multiple variables denoted by different letters, say for f .x; y/ we write,
of course,
@f .x; y/ @f .x; y/
and ; etc.
@x @y
This notation is slightly inconsistent: the xk in the “denominator” @xk just indicates
focusing on the k-th variable while the xn in the f .x1 ; : : : ; xn / in the “numerator”
refers to an actual value of the argument. When confusion is possible, one can write
more specifically
ˇ
@f .x1 ; : : : ; xn / ˇˇ
ˇ :
@xk .x1 ;:::;xn /D.a1 ;:::;an /
@.x 2 C e xyCsin.y/ /
D 2x C ye xyCsin.y/ ;
@x
@.x 2 C e xyCsin.y/ /
D .x C cos.y//e xyCsin.y/ :
@y
2.1.1
@f .x1 ; : : : ; xn /
It can happen (and typically it does) that partial derivatives exist for
@xk
all .x1 ; : : : ; xn / in some domain D 0 D. In such case, we obtain a function
@f
W D 0 ! R:
@xk
2.2
We shall write
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68 3 Multivariable Differential Calculus
for the distance of x from o (for our purposes we could have taken any of the
equivalent
p distances (recall Subsection 2.2 of Chapter 2) such as the Euclidean norm
.xx/ where xx is the dot product (see Appendix A, 4.3); our choice is perhaps the
most convenient technically because of its simple behavior with respect to products).
We say that f .x1 ; : : : ; xn / has a total differential at a point a D .a1 ; : : : ; an /
if there exists a function continuous in a neighborhood U of o which satisfies
.o/ D 0 (in an alternate but equivalent formulation, one requires to be defined
in U X fog and satisfy lim .h/ D 0), and numbers A1 ; : : : ; An such that
h!o
X
n
f .a C h/ f .a/ D Ak hk C jjhjj.h/ (2.2.1)
kD1
@f .a/
D Ak :
@xk
Proof. 1. We have
It may now seem silly to prefer the basis vectors in Rn when defining partial
derivatives. In effect, for any vector v 2 Rn , one can define a directional derivative
of f by v by
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2 Partial derivatives. Defining the existence of a total differential 69
f .x C hv/ f .x/
@v f .x/ D lim :
h!0 h
(Caution: Some calculus textbooks use a different convention, calling the @v=jjvjj
the directional derivative when v ¤ o, the point being that it only depends on the
“direction” of v. The notion as we defined it, without requiring any assumption
on v, and moreover linear in v, is much more natural for use in geometry, as we
will see later.) In any case, the following fact is proved precisely in the same way as
Proposition 2.3:
X
n
@v f .a/ D Ak vk :
kD1
2.5
The formula
X
n
f .x1 C h1 : : : ; xn C hn / f .x1 ; : : : xn / D f .a C h/ f .a/ D Ak hk C jjhjj.h/
kD1
by the required properties of , the error term is much smaller than the difference
x a.
In case of just one variable, there is no distinction between having a derivative at
a and having a total differential at the same point. In case of more than one variable,
however, the difference between having all partial derivatives and having a total
differential at a point is tremendous.
A function f may have all partial derivatives in an open set without f even
being even continuous there: In the example 1.1 (2), both partial derivatives exist
everywhere. If we consider a single point, there are even much simpler examples,
say the function f defined by f .x; 0/ D f .0; y/ D 0 for all x; y, and f .x; y/ D 1
otherwise. Then both @f @x
and @f
@y
still exist at the point .0; 0/).
What is happening geometrically is this: If we think of a function f as
represented by its “graph”, the hypersurface
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70 3 Multivariable Differential Calculus
the partial derivatives describe just the tangent lines in the directions of the
coordinate axes, while a total differential guarantees the existence of an entire
tangent hyperplane.
Possessing continuous partial derivatives is another matter, though.
Proof. Let
Set
X @f .a1 ; : : : ; ak C k hk ; : : : ; an / @f .a/ hk
.h/ D . / :
@xk @xk jjhjj
ˇ ˇ
ˇ hk ˇ
Since ˇˇ ˇ 1 and since the functions @f are continuous, lim .h/ D 0. t
u
jjhjj ˇ @xk h!o
2.7
continuous PD ) TD ) PD
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3 Composition of functions and the chain rule 71
(where PD stands for all partial derivatives and TD for total differential). Note that
neither of the implications can be reversed. We have already discussed the second
one; for the first one, recall that for functions of one variable the existence of a
derivative at a point coincides with the existence of a total differential there, but a
derivative is not necessarily a continuous function even when it exists at every point
of an open set.
In the rest of this chapter, simply assuming that partial derivatives exist will
almost never be enough. Sometimes the existence of the total differential will
suffice, but more often than not we will assume the existence of continuous partial
derivatives.
3.1 Theorem. Let f .x/ have a total differential in a point a. Let real functions
gk .t/ have derivatives at a point b and let gk .b/ D ak for all k D 1; : : : ; n. Put
X
n
@f .a/
F 0 .b/ D gk0 .b/:
@xk
kD1
1 1
.F .b C h/ F .b// D .f .g.b C h// f .g.b//
h h
1
D .f .g.b/ C .g.b C h/ g.b/// f .g.b//
h
Xn
gk .b C h/ gk .b/ jgk .b C h/ gk .b/j
D Ak C .g.b C h/ g.b// max :
h k h
kD1
1 X gk .b C h/ gk .b/
n
lim .F .b C h/ F .b// D lim Ak
h h
kD1
X
n
gk .b C h/ gk .b/ X @f .a/
n
D Ak lim D g 0 .b/: t
u
h @xk k
kD1 kD1
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72 3 Multivariable Differential Calculus
3.1.1 Corollary. Let f .x/ have a total differential at a point a. Let real functions
gk .t1 ; : : : ; tr / have partial derivatives at b D .b1 ; : : : ; br / and let gk .b/ D ak for
all k D 1; : : : ; n. Then
3.1.2 Remark
The assumption of the existence of total differential in 2.1 is essential and it is
easy to see why. Recall the geometric intuition from 2.5. The n-tuple of functions
g D .g1 ; : : : ; gn / represents a parametrized curve in D, and f ı g is then a curve
on the hypersurface S of 2.5, (*). The partial derivatives of f , or the tangent lines
of S in the directions of the coordinate axes, have in general nothing to do with the
behaviour on this curve.
The perceptive reader has noticed that in fact, while we defined what it means that
a function has a total differential, we have not yet defined the total differential
as an object. To remedy this, let us go one step further and consider in 3.1.1 a
mapping f D .f1 ; : : : ; fs /T W D ! Rs . Take its composition f ı g with a mapping
g W D 0 ! Rn (recall the convention in 1.3). Then we get
This formula is often referred to as the chain rule. It certainly has not escaped the
reader’s attention that the right-hand side is the product of matrices
@fi @gk
:
@xk i;k @xj k;j
Recall that the multiplication of matrices is the matrix of the composition of the
linear maps the matrices represent (see Theorem 7.6 of Appendix A).
In view of this, it is natural to define the total differential Dfx0 W Rn ! Rs of the
map f at a point x0 2 D as the linear map
f A W Rn ! Rs
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3 Composition of functions and the chain rule 73
For the purposes of practical calculation, in fact, the map Dfx0 and its associated
matrix A are often identified.
The chain rule can be then stated in the form
for 1
1 matrices we of course have .a/.b/ D .ab/.
Note that additionally, the total differential in this point can be used to define an
affine approximation fxaff0 of the map f at the point x0 (in an affine map approximating
f near x0 , see Appendix A, 5.9):
X
n
@f .x C .y x//
f .y/ f .x/ D .yj xj /:
j D1
@xj
X
n
@f .g.t// X
n
@f .g.t//
0
F .t/ D gj0 .t/ D .yj xj /:
j D1
@xj j D1
@xj
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74 3 Multivariable Differential Calculus
X
n
@f .x C h/
f .x C h/ f .x/ D hj :
j D1
@xj
3.4
It may be of interest that the formula for the derivative of a product of single-variable
functions is a consequence of the chain rule.
Set h.u; v/ D u v so that @f@u
D v and @f
@v
D u. Then
4.1
@g.x/
:
@xl
@2 f .x/
:
@xk @xl
More generally, we may iterate this process to obtain
@r f .x/
:
@xk1 @xk2 : : : @xkr
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4 Partial derivatives of higher order. Interchangeability 75
These functions, when they exist, are called partial derivatives of order r.
For example,
@3 f .x; y; x/ @3 f .x; y; x/
and
@x@y@z @x@x@x
are derivatives of third order (even though in the first case, we have taken a partial
derivative by each variable only once).
To simplify notation, taking partial derivatives by the same variable more than
once consecutively may be indicated by an exponent, e.g.,
@5 f .x; y/ @5 f .x; y/
2 3
D ;
@x @y @x@x@x@y@y
@5 f .x; y/ @5 f .x; y/
2 2
D :
@x @y @x @x@x@y@y@x
4.2
Compute
@f .x; y/ @f .x; y/
D sin.y 2 C x/ C x cos.y 2 C x/ and D 2xy cos.y 2 C x/:
@x @y
@2 f @2 f
D 2y cos.y 2 C x/ 2xy sin.y 2 C x/ D :
@x@y @y@x
@2 f
4.2.1 Proposition. Let f .x; y/ be a function such that the partial derivatives
@x@y
@2 f
and are defined and continuous in a neighborhood of a point .x; y/. Then we
@y@x
have
@2 f .x; y/ @2 f .x; y/
D :
@x@y @y@x
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76 3 Multivariable Differential Calculus
f .x C h; y C h/ f .x; y C h/ f .x C h; y/ C f .x; y/
F .h/ D :
h2
If we set
we have
1 1
F .h/ D .'h .y C h/ 'h .y// D 2 . h .x C h/ h .x//:
h2 h
Let us compute the first expression. The function 'h , which is a function of one
variable y, has the derivative
@f .x C h; y/ @f .x; y/
'h0 .y/ D
@y @y
1 1
F .h/ D 2
.'h .y C h/ 'h .y// D 'h0 .y C 1 h/
h h
@f .x C h; y C 1 h/ @f .x; y C 1 h/
D :
@y @y
@ @f @ @f
Now since both . / and . / are continuous at the point .x; y/, we can
@y @x @x @y
compute lim F .h/ from either of the formulas (*) or (**) and obtain
h!0
@2 f .x; y/ @2 f .x; y/
lim F .h/ D D : t
u
h!0 @x@y @y@x
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5 The Implicit Functions Theorem I: The case of a single equation 77
Remark. Look what happens: F .h/ (and its possible limit in 0) is an attempt
@2 f
to compute the second partial derivative in one step. The continuity of and
@x@y
@2 f
makes sure that it is, in fact, possible.
@y@x
4.3
4.3.1
Thus, under the assumption of the theorem, we can write a general partial derivative
of the order r k as
@r f
with r1 C r2 C C rn D r
@x1r1 @x2r2 : : : @xnrn
where, of course, rj D 0 is allowed and indicates the absence of the symbol @xj .
5.1
F .x; y/;
F .x; y/ D 0: (5.1.1)
Even in very simple cases we can hardly expect a unique solution. Take for example
F .x; y/ D x 2 C y 2 1. Then for jxj > 1 there is no solution f .x; y/. For jx0 j < 1,
for some open interval containing x0 , we have two solutions
p p
f .x/ D 1 x2 and g.x/ D 1 x 2 :
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78 3 Multivariable Differential Calculus
This is better, but we have two values in each point, contradicting the definition of
a function. To achieve uniqueness, we have to restrict not only the values of x, but
also the values of y to an interval .y0
; y0 C
/ (where F .x0 ; y0 / D 0). That is,
if we have a particular solution .x0 ; y0 / we must restrict our attention to a “window”
.x0 ı; x0 C ı/
.y0
; y0 C
/
y 2 jxj D 0;
the solution .0; 0/ can be extended indefinitely both ways, but still there is no
neighborhood of .0; 0/ in which there would be a unique solution.
5.2
Actually, the above examples cover more or less all the exceptions that can occur
for “reasonable” functions F .
Then there exist ı > 0 and
> 0 such that for every x with jjx x0 jj < ı there
exists precisely one y with jy y0 j <
such that
F .x; y/ D 0:
Furthermore, if we write y D f .x/ for this unique value y, then the function
f W .x10 ı; x10 C ı/
.xn0 ı; xn0 C ı/ ! R
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5 The Implicit Functions Theorem I: The case of a single equation 79
@F .x0 ; y0 /
> 0:
@y
Since the first partial derivatives of F are continuous, there exist a > 0, K, ı1 > 0
and
> 0 such that for all .x; y/ 2 J.ı1 /
hy0
; y0 C
i, we have
ˇ ˇ
@F .x; y/ ˇ @F .x; y/ ˇ
a and ˇ ˇK (5.2.1)
@y ˇ @x ˇ
i
By 2.6 and 2.3, F is continuous, and hence there is a ı, 0 < ı ı1 , such that
x D .xb ; xj ; xa /:
@f
Compute @x j
as the derivative of .t/ D f .xb ; t; xa /.
By 3.3, we have
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80 3 Multivariable Differential Calculus
and hence
and f is continuous (note that we have not known that before). Using this fact,
we can compute from (5.2.2)
.t C h/ .t /
lim D
h!0 h
@F .xb ; t C h; xa ; .t / C . .t C h/ .t /// @F .xb ; t; xa ; .t //
@xj @xj
D lim D :
h!0 @F .xb ; t C h; xa ; .t / C . .t C h/ .t /// @F .xb ; t; xa ; .t //
@y @y
III The higher derivatives. Note that we have not only proved the existence of the
first derivative of f , but also the formula
1
@f .x/ @F .x; f .x// @F .x; f .x//
D : (5.2.3)
@xj @xj @y
From this we can inductively compute the higher derivatives of f (using the
standard rules of differentiation) as long as the derivatives
@r F
@x1r1 @xnrn @y rnC1
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6 The Implicit Functions Theorem II: The case of several equations 81
5.3
We have obtained the formula (5.2.3) while proving that f has a derivative. If we
knew beforehand that f has a derivative, we could deduce (5.2.3) immediately from
the chain rule. In effect, we have
0 F .x; f .x//I
5.4 Remark
The solution f in 5.2 has as many derivatives as the initial F . But note the restriction
r 1. One usually thinks of the 0-th derivative as of the function itself. The theorem
does not guarantee a continuous solution f of an equation F .x; f .x// D 0 with
continuous F . Even just for the existence of the f we have used the first derivatives.
F1 .x; y1 ; y2 / D 0;
F2 .x; y1 ; y2 / D 0
in a neighborhood of a point .x0 ; y10 ; y20 / (at which the equalities hold). We will
apply the “substitution method” based on Theorem 5.2. First we will think of
the second equation as an equation for the unknown y2 ; in a neighborhood of
.x0 ; y10 ; y20 / we obtain y2 as a function .x; y1 /. Substitute this into the first equation
to obtain
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82 3 Multivariable Differential Calculus
What did we have to assume? First, of course, we have to have the continuous
partial derivatives of the functions Fi . Then, to be able to obtain by 5.2 the way
we did, we need to have
@F2 0 0 0
.x ; y1 ; y2 / ¤ 0: (6.1.1)
@y2
Finally, we also need to have
@G 0 0
.x ; y1 / ¤ 0I
@y1
by 3.1.1, this is equivalent to
@F1 @F1 @
C ¤ 0: (6.1.2)
@y1 @y2 @y1
Now we have (recall (5.2.3))
@ @F1 1 @F2
D
@y1 @y2 @y1
and (6.1.2) becomes
1
@F1 @F1 @F2 @F1 @F2
¤ 0;
@y2 @y1 @y2 @y2 @y1
that is,
This formula should be conspicuously familiar. Indeed, it is (see the notation for
determinants from Subsection 3.3 of Appendix B)
ˇ ˇ
ˇ @F1 @F1 ˇ
ˇ ˇ
ˇ @y1 ; @y2 ˇ
ˇ ˇ @Fi
ˇ ˇ D det ¤ 0: (6.1.3)
ˇ @F2 @F2 ˇˇ @yj i;j
ˇ
ˇ @y ; @y2 ˇ
1
@F2 0 0 0
.x ; y1 ; y2 / ¤ 0;
@y1
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6 The Implicit Functions Theorem II: The case of several equations 83
6.3
By extending the substitution procedure indicated in 6.1, we will now prove the
general Implicit Function Theorem.
F.x0 ; y0 / D o
and let
D.F/ 0 0
.x ; y / ¤ 0:
.y/
Then there exist ı > 0 and > 0 such that for every
x 2 .x10 ı; x10 C ı/
.xn0 ı; xn0 C ı/
y 2 .y10
; y10 C
/
.ym
0
; xm
0
C
/
such that
F.x; y/ D 0:
Furthermore, if we write this y as a vector function f.x/ D .f1 .x/; : : : ; fm .x//, then
the functions fi have continuous partial derivatives up to the order k.
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84 3 Multivariable Differential Calculus
Fi .x; y/; i D 1; : : : ; m C 1
@FmC1 0 0
.x ; y / ¤ 0:
@ymC1
This has continuous partial derivatives up to the order k and hence so have the
functions
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6 The Implicit Functions Theorem II: The case of several equations 85
@
Add to the i th column the product of the last column with the scalar . By (6.3.1),
@yi
taking into account the fact that GmC1 0 and hence
we obtain
ˇ ˇ
ˇ @G1 @G1 @F1 ˇ
ˇ ˇ
ˇ @y1 ; : : : ; ;
@ym @ymC1 ˇ
ˇ ˇ
ˇ ˇ
ˇ ˇ
ˇ :::; :::; :::; ::: ˇ
ˇ ˇ
ˇ ˇ
D.F/ ˇ ˇ @FmC1 D.G1 ; : : : ; Gm /
Dˇ ˇD :
D.y/ ˇ @Gm @Gm @Fm ˇ @ymC1 D.y1 ; : : : ; ym /
ˇ ˇ
ˇ @y ; : : : ; ; ˇ
ˇ 1 @ym @ymC1 ˇ
ˇ ˇ
ˇ ˇ
ˇ @FmC1 ˇ
ˇ 0; : : : ; 0; ˇ
ˇ @ymC1 ˇ
Thus,
D.G1 ; : : : ; Gm /
¤0
D.y1 ; : : : ; ym /
and hence by the induction hypthesis there are ı2 > 0,
2 > 0 such that for
jxi xi0 j < ı2 there is a uniquely determined yQ with jyi yi0 j <
2 such that
Gi .x; yQ / D 0 for i D 1; : : : ; m
and that the resulting fi .x/ have continuous partial derivatives up to the order k.
If we define, further,
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86 3 Multivariable Differential Calculus
We have to prove that then necessarily yi D fi .x/ for all i . Since jxi xi0 j < ı ı1
for i D 1; : : : ; n, jyi yi0 j <
ı1 for i D 1; : : : ; m and jymC1 ymC1
0
j <
1
we have, necessarily, ymC1 D .x; yQ /. Thus, by (6.3.2),
G.x; yQ / D o
and since jxi xi0 j < ı ı2 and jyi yi0 j <
2 we have indeed yi D fi .x/.
t
u
7.1
D.f/
.x/ ¤ 0:
D.x/
.y0 ; ı/ D fy j jjy y0 jj < ıg f ŒV : t
u
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8 Taylor’s Theorem, Local Extremes and Extremes with Constraints 87
Proof. Consider the F from (7.2.1) again. We have, for a sufficiently small
> 0,
precisely one x D g.y/ such that F.x; y/ D 0 and jjx x0 jj <
. This g has,
furthermore, continuous partial derivatives. We have, by 3.2,
D.f/ D.g/
D detDf detDg D 1
D.x/ D.y/
D.g/
and hence for each y 2 fŒV , D.y/
.y/ ¤ 0. t
u
f .x/ D
X
r X 1 @k f .a/
.x1 a1 /k1 : : : .xn an /kn
k1 Š : : : kn Š .@x1 / 1 : : : .@xn /kn
k
kD0 k1 CCkn Dk; ki 0
X 1 @k f .c/
C .x1 a1 /k1 : : : .xn an /kn :
k1 Š : : : kn Š .@x1 / 1 : : : .@xn /kn
k
k1 CCkn DrC1; ki 0
(*)
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88 3 Multivariable Differential Calculus
g .k/ .t/ D
ˇ
X @k f .s/ ˇ
kŠ ˇ
ˇ .x1 a1 /k1 : : : .xn an /kn
k1 Š : : : kn Š .@s1 /k1 : : : .@sn /kn ˇ
k1 CCkn Dk ki 0 sD.aCt .xa//
(**)
It is useful to note that the affine approximation in the sense of 3.2 of the function
f at a point a is simply the sum of the constant and linear terms of its Taylor
expansion.
@2 f
:
@xi @xj
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8 Taylor’s Theorem, Local Extremes and Extremes with Constraints 89
This is a symmetric matrix by Proposition 4.2.1, and hence has an associated real
symmetric bilinear form. If the Hessian is non-degenerate at a critical point x0 , we
call x0 a non-degenerate critical point. We have the following
1
f .x/ D f .x0 / C .x x0 /T H.c/.x x0 /: (8.3.1)
2
.v; c/ D vT H.c/v
X D f.v; c/ j v 2 Rn ; v v D 1; c 2
.x0 ; =2/g:
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90 3 Multivariable Differential Calculus
@f @f
D 0; D 0: (*)
@x @y
On the boundary, the extreme may not satisfy the equations (*), but we note that the
boundary is itself the set of solutions of the “nice” equation
x 2 C y 2 D 1: (C)
It is certainly worth asking if some generalization of (*) might hold, which would
allow us to solve the problem. Note that generically speaking, we expect a single
equation in the boundary case, since in addition to it, we still have the equation
(“constraint”) (C).
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8 Taylor’s Theorem, Local Extremes and Extremes with Constraints 91
gi .x/ D 0; i D 1; : : : ; k (*)
if x D x0 satisfies (*) and there exists a ı > 0 such that for every x 2
.x0 ; ı/,
x ¤ x0 which satisfies (*) we have f .x/ > f .x0 / (resp. f .x/ < f .x0 /.
We have the following
@f .a/ X
n
@gj .a/
C j D 0:
@xi j D1
@xi
Proof. See Subsection 2.4 of Appendix B. If the matrix M has rank k, then at least
one of the k
k submatrices of M is regular, and hence has a non-zero determinant.
Without loss of generality, let us assume that at the extremal point we have, say,
ˇ ˇ
ˇ @g1 @g1 ˇ
ˇ ˇ
ˇ @x1 ; : : : ; @xn ˇ
ˇ ˇ
ˇ ˇ
ˇ ˇ
ˇ :::; :::; : : : ˇˇ ¤ 0:
ˇ (1)
ˇ ˇ
ˇ ˇ
ˇ ˇ
ˇ @gk @gk ˇˇ
ˇ
ˇ @x ; : : : ; @xn ˇ
1
i .xkC1 ; : : : ; xn /
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92 3 Multivariable Differential Calculus
(let us write xQ for .xkC1 ; : : : ; xn /) with contiuous partial derivatives such that
Thus, an extreme (i.e. local maximum or local minimum) of f .x/ at a subject to the
given constraints implies the corresponding extreme property (without constraints)
of the function
@F .Qa/
D0 for i D k C 1; : : : ; n;
@xi
X
k
@f .a/ @r .Qa/ @f .a/
C for i D k C 1; : : : ; n: (2)
rD1
@xr @xi @xi
X
k
@gj .a/ @r .Qa/ @gj .a/
C for i D k C 1; : : : ; n: (3)
rD1
@xr @xi @xi
Now we will use (1) again, for another purpose. By Theorem B.2.5.1, the system of
linear equations
@f .a/ X
n
@gj .a/
C j D 0; i D 1; : : : ; k;
@xi j D1
@xi
has a unique solution 1 ; : : : ; k . Those are the equalities from the statement, but,
so far, for i k only. It remains to be shown that the same equalities hold also for
i > k. In effect, by (2) and (3), for i > k we obtain
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8 Taylor’s Theorem, Local Extremes and Extremes with Constraints 93
8.6 Remarks
8.7
V .x1 ; : : : ; xn / D x1 xn :
Thus, we have
@V 1 @S 2 1 1 1
D x1 xn and D .x1 xn / CC 2x1 xn 2 :
@xi xi @xi xi x1 xn xi
If we write yi D x1i and s D y1 C C yn and divide the equation from the theorem
by x1 xn , we obtain
2yi .s yi / C yi D 0; or yi D s C :
2
Thus, all the xi are equal and the unique solution is the cube.
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94 3 Multivariable Differential Calculus
9 Exercises
x y
f .x; y/ D e y x for x; y ¤ 0
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9 Exercises 95
(12) (a) Find a maximum and minimum of the function f .x; y/ D ax C by on the
set
B D f.x; y/ j x 2 C y 2 1g R2
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In the first part of this chapter we will present a simple generalization of the one-
dimensional Riemann integral which the reader already knows (see Section 8 of
Chapter 1). To start with, we will consider the integral only for functions defined
on n-dimensional intervals (D“bricks”) and we will be concerned, basically, with
continuous functions. Later, the domains and functions to be integrated on will
become much more general.
1.1
J D ha1 ; b1 i
han ; bn i
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98 4 Integration I: Multivariable Riemann Integral and Basic Ideas Toward the: : :
1.2
Y
n
volJ D .bi ai /:
i D1
where
1.2.1
From the definitions of suprema and infima we immediately see that if D refines D 0
then
s.f; D/ S.f; D 0 /:
Now we can define the lower and the upper Riemann integral of f over J by
setting
Z Z
f D sup s.f:D/ and f D inf S.f:D/;
D J D
J
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1 Riemann integral on an n-dimensional interval 99
and if these two values coincide we speak of the Riemann integral of f over J and
write
Z
f
J
1.3
The following easy fact can be left to the reader (it can be proved by a literal
repetition of the one variable case – Exercise (1)).
An interval J D ha1 ; b1 i
han ; bn i is an almost disjoint union of a pair of
intervals J i D ha1i ; b1i i
hani ; bni i, i D 1; 2, if for some k we have
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100 4 Integration I: Multivariable Riemann Integral and Basic Ideas Toward the: : :
Proof. It suffices to prove the statement for an almost disjoint union of a pair of
intervals J 1 ; J 2 , and for this case it suffices to realize that each partition of J can
be refined into a pair of partitions of the J i ’s, and, on the other hand, from any pair
of partitions of the J i ’s we can obtain, using common refinements, a partition of J .
t
u
2.1 Theorem. A function F is Riemann integrable if and only if for every " > 0
there exists a partition D such that
R R
On the other hand, if f D J f then by definition there are D 0 ; D 00 such that
J
S.D; f / s.D; f / < "; take a common refinement D of D 0 ; D 00 and use 1.2.1 (*).
t
u
Proof. By Theorem 6.6 of Chapter 2, f is uniformly continuous. Take an " > 0 and
choose a ı > 0 such that for the distance in Rn we have
"
d.x; y/ < ı ) jf .x/ f .y/j < :
volJ
Further, choose a partition D such that
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3 Fubini’s Theorem in the continuous case 101
we have
X " X
S.D; f / s.D; f / D .MK mK /volK volK D ": t
u
volJ
K2jDj K2jDj
2.2
The following statements are straightforward (they hold more generally, but we will
need them so far for continuous functions only).
Proposition.
R R Let f; g be continuous functions. Then
1. j f j jf j.R R
2. If f g then f g.
3. In particular if f .x/ C for all x 2 J then
Z
f C volJ:
J
Proof. We will Rprove the first equality, the second one is analogous.
Put F .x/ D J 00 f .x; y/dy. We will prove that
Z Z
f D F:
j j0
This will also include the fact that the latter integral exists; this could be easily
shown by proving, using uniform continuity, that F is continuous. But we will get
it during the proof for free anyway.
Choose a partition D of J such that
Z Z
f " s.f; D/ S.f; D/ f C ":
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102 4 Integration I: Multivariable Riemann Integral and Basic Ideas Toward the: : :
jDj D fK 0
K 00 j K 0 2 jD 0 j; K 00 2 jD 00 jg
and hence
X X
S.F; D 0 / max0 . max f .x; y/ volK 00 / volK 0
x2K y2K 00
K 0 2jD 0 j K 00 2jD 00 j
X X
max f .x; y/ volK 00 volK 0
.x;y/2K 0 K 00
K 0 2jD 0 j K 00 2jD 00 j
X
max f .z/ vol.K 00
K 0 / D S.f; D/
z2K 0 K 00
K 0 K 00 2jDj
and similarly
s.f; D/ s.F; D 0 /:
Hence we have
Z Z Z Z
0
f " s.F; D / S.F; D/ f C ";
j J0 J0 J
R R
and therefore J0 F exists and is equal to J f. t
u
4.1 Theorem. Let fn be continuous real functions on a compact interval J and let
them converge uniformly to a function f . Then
Z Z
f D lim fn :
J n!1 J
The symbols mK and MK will be as in 1.2, and the corresponding values for fn will
be denoted by mnK and MKn . Thus we have
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4 Uniform convergence and Dini’s Theorem 103
"
jmK mnk j; jMK Mkn j <
volJ
so that
X
js.f; D/ s.fn ; D/j jmK mnK j volK < "
K2jDj
X
(again we use the fact that volK D volJ ) and similarly
K2jDj
Then
Z Z
f 2" s.f; D/ " s.fn ; D/ fn
J J
Z
S.fn ; D/ S.f; D/ C " f C 2";
J
R R
and we conclude that lim J fn D J f. t
u
4.2 Notation
fn % f resp. fn & f
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104 4 Integration I: Multivariable Riemann Integral and Basic Ideas Toward the: : :
Proof. It suffices to prove that mn D max fn .x/ converges to zero, because then
x
jfn .x/ 0j < " for sufficiently large n independently of the choice of x 2 X .
Suppose it does not. Reducing, possibly, fn to a subsequence, we obtain an
example with
Now for k n,
and hence
4.4
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5 Preparing for an extension of the Riemann integral 105
5.1
For many purposes, the Riemann integral is not sufficiently general. For example,
we may be interested in computing integrals such as
Z 1
dx p
p D 2 xj10 D 2;
0 x
which however is incorrect in the setting we considered so far, since the Riemann
integral on the left-hand side does not exist. While in this particular case there is a
quick fix in the form of “improper Riemann integrals” (which we do not treat here),
clearly, a more systematic solution is needed: What about a function f where f .x/
is 0 for x rational and 1 for x irrational? (This function is known as the Dirichlet
function.) Obviously, f is not Riemann integrable, but should we define
Z 1
f .x/dx D 1
0
to express that modifying the value of the function which is constantly equal to 1 on
countably many points should not change the value of the integral? More generally,
can one define the integral in such a way that we have
Z Z
lim fn D lim fn (*)
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106 4 Integration I: Multivariable Riemann Integral and Basic Ideas Toward the: : :
supp.f /:
Thus, a function has compact support if and only if it vanishes outside a compact
subset X Rn . There is obviously the smallest interval J0 containing the set X .
Any interval J containing J0 is easily represented as an almost disjoint union of a
set of intervals containing J0 and Rsuch that f is zero on all the other members of the
system. Thus by 1.4, the integral J f does not depend on the choice of the interval
J containing the support X of f . We will denote the common value by
If
R
(we will reserve the standard symbol for an extended integral defined later).
The set of all continuous functions with compact support in Rn will be denoted by
Z:
Let us summarize the basic facts we will use below: We have a class Z of
functions defined on Rn such that
(Z1) for all ˛; ˇ 2 R and f; g 2 Z, ˛f C ˇg 2 Z,
(Z2) if f 2 Z then jf j 2 Z,
and a mapping I W Z ! R such that
(I1) if f 0 then If 0,
(I2) I is a linear map, and
(I3) if fn & 0 then Ifn & 0
(for (I3), use 4.4, realizing that the support of fn is contained in the support of f1 ).
Below, we will consistently use only the facts (Zj) and (Ij) and their conse-
quences. For example, let max.f; g/ (resp. min.f; g/) denote the function whose
value at a point x is max.f .x/; g.x// (resp. min.f .x/; g.x//), and let f C D
max.f; 0/, f D min.f; 0/. Note that
1 1
max.f; g/ D .f C g C jf gj/ and min.f; g/ D .f C g jf gj/:
2 2
Thus, we easily deduce that
f g ) If Ig; and
f; g 2 Z ) max.f; g/; min.f; g/; f C ; f 2 Z:
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6 A modest extension 107
6 A modest extension
6.1
Define
Zup D ff W Rn ! .1; C1 j 9fn 2 Z; fn % f g;
Zdn D ff W Rn ! Œ1; 1/ j 9fn 2 Z; fn & f g;
Z D Zup [ Zdn :
hn D min.gn ; fk /:
lim hn D min.g; fk / D fk ;
and hence
and we obtain, by (I3), that lim Ihn D Ifk . Now gn hn , hence Ign Ihn ,
n
and hence
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108 4 Integration I: Multivariable Riemann Integral and Basic Ideas Toward the: : :
If D lim Ifn
n
gn D maxffij j 1 i; j ng:
we have
gn fn f: (1)
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7 A definition of the Lebesgue integral and an important lemma 109
g fn : (2)
In this section, we will define the well-known Lebesgue integral by the method of
Daniell. This approach differs from the original Lebesgue construction based on
defining a measure first. Here we will obtain measure later as a consequence of an
already defined integral. We will see in Chapter 5 that the basic properties of
measure will follow practically for free.
7.1
R R
f resp f is called the lower resp. upper (Lebesgue) integral of f .
Remark. This notation will not interfere with the notation for the lower and
upper Riemann integral introduced in 1.2 and used through Section 4. While the
meanings of both notations are in fact different, we will not encounter the lower and
upper Riemann integral any longer (with the exception of the Exercises).
R R
7.2 Proposition. (1) f D supfIg j g f; g 2 Z g and f D inffIg j g
f; g 2 Z g.
R R
(2) f f .
R R R R
(3) If f g then f g and f g.
Proof. (a) Assume that, say, the second equality does not hold. Then there exists a
R
g f , g 2 Zdn such that Ig < f . Let gn & g with gn 2 Z. Then there has to
R
be a k such that Igk < f . This is a contradiction, since gn 2 Z Zup .
(2) and (3) are trivial. t
u
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110 4 Integration I: Multivariable Riemann Integral and Basic Ideas Toward the: : :
7.3
7.4
Denote by
R R
the set of all functions f such that f D f and such that the common value is
finite. Such functions are called (Lebesgue) integrable, the common finite value is
called the Lebesgue integral of f and denoted by
Z
f:
We will keep this notation for a while to distinguish the Lebesgue integral from the
types of integral developed earlier. Note, however, that in practice, other notations
are also common, for example, if x1 ; : : : ; xn are the standard coordinates in Rn , one
commonly writes
Z
f .x1 ; : : : ; xn /dx1 : : : dxn
or
Z
f .x/dx
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7 A definition of the Lebesgue integral and an important lemma 111
R R
so that f f is smaller than any " > 0. t
u
7.6 Convention
Functions from L can have infinite values. Let us agree that in case of f .x/ D C1
and g.x/ D 1 the value f .x/ C g.x/ will be chosen arbitrarily. We will see that
for our purposes such arbitrariness in the definition of f C g does not matter.
Proof. (1) We shall use 7.5. Choose f1 ; g1 2 Zup and f2 ; g2 2 Zdn such that f1
f f2 , g1 g g2 and If1 If2 < ", Ig1 Ig2 < ". Then
f1 C g1 f C g f2 C g2 (*)
and the statement follows (realize that the inequalities hold also at the ambigu-
ous points mentioned in the convention of 7.6: if, say, f .x/ D C1 and
g.x/ D 1 then f2 .x/ D C1 and g1 .x/ D 1; f1 .x/ has to be finite,
as a limit of a decreasing sequence of finite numbers, and similarly for g2 .x/ so
that the inequalities (*) are satisfied trivially).
(2) follows immediately from 7.5.
(3) Take the fi ; gi as in (1) to obtain
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112 4 Integration I: Multivariable Riemann Integral and Basic Ideas Toward the: : :
R R R
Proof. We obviously have lim fn f , and if lim fn D C1 the equality is
trivial. R
Thus, we can assume that the limit is finite. By the definition of fn choose
gn 2 Zup , gn fn such that
Z
"
fn C > Ign :
2nC1
(Indeed, at each point x, we have gj .x/ fj .x/ D hn .x/ fj .x/ for some
j n. The summands are non-negative, and hence the inequality holds for j D n;
otherwise the sum is greater than or equal to hn .x/ fj .x/ C gn .x/ fn .x/ D
hn .x/fn .x/Cgn .x/fj .x/ hn .x/fn .x/Cgn .x/fn .x/ hn .x/fn .x/.)
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8 Sets of measure zero; the concept of “almost everywhere” 113
Thus we have
Z X n
"
Ihn fn i C1
<"
i D1
2
R R R
so that Ihn fn C " and finally f Ihn lim fn C ". t
u
Set
Lup \ Ldn D L:
7.9.2 Convention R R R
For f 2 L we will use the symbol f for the common value of f and f , even
when it is infinite. However, we will not refer to such functions as integrable.
R
7.9.3 Proposition. If f 2 L and if the integral f from 7.9.2 is finite then f 2 L
and the integral coincides with the standard integral in L.
Proof. Let, say, f 2 Lup , let fn % f with fn 2 L. Then by Lemma 7.8 and part 2
R R R R
of the Remark in 7.8, f D lim fn D f D f . t
u
8.1
cM
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114 4 Integration I: Multivariable Riemann Integral and Basic Ideas Toward the: : :
M N if and only if cM cN ;
cM [N D max.cM ; cN / and cM \N D min.cM ; cN /;
S
1
and if M1 M2 Mn , M D Mn , then
nD1
cMn % cM :
R
R M is a set of measure zero if cM D 0 (then, since cm 0, we also have
cM D 0 and hence cM 2 L).
Proof. (1) is trivial. For (2), consider Nn D M1 [ [Mn . Then cNn cM1 C cMn
R
and hence Nn is a set of measure zero by 7.7. Now cNn % cM and hence cM D 0
by 7.8. t
u
8.3
fx j not V .x/g
f g:
Proof. (1) Recall the convention on sums in 7.6, and Proposition 7.7 (1). We may
define f CR .f / equally
R well as 0 or as cM where M D fx j f .x/ D ˙1g
and hence cM D 0 D 0.
(2) When f 2 Lup , take fn 2 L with fn % f . Then fx j f .x/ D 1g
fx j f1 .x/ D ˙1g and the latter set is a set of measure zero by (1). The case
of f 2 Ldn is analogous. t
u
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9 Exercises 115
R R R R
8.5 Proposition. If f
g then f D g and f D g.
R
Proof. We will consider the case of (the other case is analogous). If we do not
R R R
have f D g D C 1 we can assume that f < C 1. Set M Dfxj f .x/ ¤ g.x/g
R
and rn D n cM . By 3.8 we have r D 0 for r D lim rn .
R
Choose h1 ; h2 2 Zup such that h1 f , h2 r, Ih1 < f C " and Ih2 < ".
R R
Then we have h1 C h2 2 Zup , h1 C h2 g, and hence g Ih1 C Ih2 < f C 2".
R R R
Thus, g f , in particular g < C1, and we can repeat the procedure with
f; g interchanged. t
u
9 Exercises
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116 4 Integration I: Multivariable Riemann Integral and Basic Ideas Toward the: : :
T
(9) By a generalized Cantor set, we shall mean the intersection S D Si of sets
S0 S1 S2 : : : constructed as follows: We put S0 D h0; 1i. The set Sn
is a union of 2n closed intervals hai ; bi i, i D 1; : : : ; 2n , and for some number
bi ai
"n > 0, " < , we have
2
2n
!
[ ai C bi ai C bi
SnC1 D Sn X . "; C" :
i D1
2 2
(a) Prove that there exist generalized Cantor sets which are not of measure 0.
(b) Derive a necessary and sufficient condition (in terms of the numbers "i )
for the set S to be of measure 0.
(10) (a) Prove that for two generalized Cantor sets S , T , there exists a monotone
homeomorphism W h0; 1i ! h0; 1i such that ŒS D T . [Hint: Construct
such map with S , T replaced by Sn , Tn and prove that the sequence of
those maps converges uniformly. Use a separate argument to show that
the limit is monotone.]
(b) Conclude that for a homeomorphism h0; 1i ! h0; 1i, a continuous image
of a set of measure 0 may not be of measure 0.
(11) Let f W R ! h0; 1i be defined as follows: If x is irrational, then f .x/ D 0. If
x D a=b where a 2 Z, b 2 N and the greatest common divisor of a and b is
1, then f .a=b/ D 1=b. Prove that f is continuous almost everywhere. [Hint:
Try to guess the set of all points at which f is continuous.]
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1 Lebesgue’s Theorems
fn & f .
Proof. Let us treat the case fn % f , fn 2 Lup , the other case is analogous. Choose
fnk 2 L such that fnk %k fn and set
gn D maxffij j i; j ng:
Remark. The attentive reader may worry about the seemingly sloppy formula-
tion: does one mean “almost everywhere one has that for all n that jfn .x/j g.x/”
or “for each n one has that jfn .x/j g.x/ almost everywhere”? But it is an easy
exercise (Exercise (1)) to show these two statements are equivalent.
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118 5 Integration II: Measurable Functions, Measure and the Techniques: : :
Proof. By 8.5 of Chapter 4, we may omit “almost everywhere” from the assump-
tions.
Set
Since max fnCj %p hn we have hn 2 Lup , and similarly gn 2 Ldn . But we have,
j D0;:::;p
moreover,
g gn fn hn g
R R
and hence gn and hn are finite and we have in fact gn ; hn 2 L, and consequently
gn 2 Lup and hn 2 Ldn and we can use Lebesgue’s Monotone Convergence
Theorem. Now obviously gn % fR and hn &R f , by RLebesgue’s Monotone
Convergence Theorem we haveR lim gn D R lim hn D f , and finally since
gn fn hn we conclude that f D lim fn . t
u
1.3 Proposition. Let g 2 L, let fn 2 L , let fn g a.e. and let lim fn .x/ D f .x/
n
a.e. Then f 2 Lup . Similarly for fn g we obtain f 2 Ldn .
R R
Proof. Since 1 < g fn , fn 2 Lup (if fn 2 Ldn it has, hence, a finite
integral so that, by 7.9.3 of Chapter 4, fn 2 L Lup as well). Set ' D supn fn .
We have max fk %n ' and hence ' 2 Lup by 1.1, and there exist 'n 2 L such that
kn
'n % '. Obviously ' f g and we can assume that 'n g (else replace 'n by
max.'n ; g/). Set
gkn D min.'k ; fn /:
We have g gkn 'k and hence gkn 2 L and, moreover, we can use Lebesgue’s
Dominated Convergence Theorem for lim gkn and obtain
n
2.1
ƒ D ff j 9fn 2 L; fn ! f g
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2 The class ƒ (measurable functions) 119
(unlike in the definition of Lup and Ldn there is no assumption on the nature of the
convergence). Functions which belong to ƒ are called (Lebesgue) measurable.
2.3
2.4
2.5.1 Corollary. Let f 2 ƒ and let there exist a g 2 L such that jf j g. Then
f 2 L.
Proof. We have fnC ; fn 2 Lup and fnC ! f C , fn ! f . Thus, by 1.3, both f C
and f are in Lup . t
u
2.7
R Proposition.
R f 2 L if and only if f C and f are in Lup and if the difference
f C f makes sense.
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120 5 Integration II: Measurable Functions, Measure and the Techniques: : :
R
R Consequently, f 2 ƒ X L if and only if f C and f are in Lup and fC D
f D C1.
Proof. ) : Let, say, f 2 Lup and let fn % f and fn 2 L. As Rf1 D f1C f1
f DfC
R f C weRhave f f1 2 L and hence the value of f is finite.
( : If f f makes sense then at least one of the integrals is finite and
either f C or f is in L. Thus, f C f is either in Lup or in Ldn . t
u
2.8 Remark
Some of the statements proved in this section may be somewhat surprising. It turned
out, for example, that for integrability of a limit of integrable functions, the nature
of the limiting process is not very important: all one needs is that the positive and
negative parts of the limit not both have infinite integrals.
For the value of the integral of the limit, on the other hand, the nature of the
convergence obviously matters a great deal.
3.1
as then cA[B D cA C cB .
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3 The Lebesgue measure 121
But we have much more: the measure is countably additive (-additive, as this
fact is usually referred to). Here are some facts on measurability.
1
[
Proposition. (1) Let An , n D 1; 2; : : : , be measurable sets. Then An is
nD1
measurable. If for any two n; k the intersection An \ Ak is a set of measure
zero then
1
[ 1
X
. An / D .An /:
nD1 nD1
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122 5 Integration II: Measurable Functions, Measure and the Techniques: : :
1
An D fx j .x; Rm X U / g
n
and define fn W Rm ! R by
.x; Rm X U /
fn .x/ D :
.x; Rm X U / C .x; An /
fn ! cU
and cU 2 ƒ.
(2) Use (1) and 3.2 (3).
(3) Note that for a bounded closed set C we can use a similar procedure as in (1):
this time set
1
An D fx j .x; C / g
n
and define fn W Rm ! R by
.x; An /
fn .x/ D :
.x; An / C .x; C /
fn & cC :
1 1 1 1
ha1 ; b1 C i
ham ; bm C i
n n n n
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4 The integral over a set 123
The smallest class of subsets of Rm containing all open subsets and closed under
• Complements,
• Countable unions, and
• Countable intersections
(of course, the last follows from the first two) is called the class of Borel sets, and
denoted by B.
Thus, all the open and closed sets are Borel. However, we have more complicated
sets. For example, an F set is a countable union of closed subsets, and a Gı set is
an intersection of countably many open subsets. Going on, a Gı set is a union of
countably many Gı sets, and an F ı set is an intersection of countably many F
sets, and so on. All sets produced in this way are Borel by definition.
From 3.2 and 3.3 we immediately obtain
3.5
Let us conclude this section with a trivial remark. From 3.2 (1) and 2.2 (1), we
immediately obtain the frequently used somewhat paradoxical observation that for
every " > 0, there exists a dense open set U of the unit interval I such that
.U / < ": order all the rationals in I in a sequence r1 ; r2 ; : : : ; rn ; : : : and set
1
[ 1 1
U D .rn ; rn C nC2 /
nD1
2nC2 2
4.1
Unlike the additivity of the classes L etc., we do not have similarly well behaved
multiplicativity properties. Nevertheless, multiplying by characteristic functions cM
of M measurable does give satisfactory results.
Proof. Put 'n D min.ncM ; .max.f; .ncM ////. Then 'n 2 ƒ and since j'n j jf j
we have cM f D lim 'n in L by 2.5.1. t
u
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124 5 Integration II: Measurable Functions, Measure and the Techniques: : :
4.2
S R
(b) Let M1 M2 ; M D Mn and assume that M f makes sense. Then
Z Z
f D lim f:
M n Mn
T R
(c) Let M1 M2 ; M D Mn and assume that M1 f makes sense. Then
Z Z
f D lim f:
M n Mn
the last reshuffling being made possible by the absolute convergence of at least one
of the series (and the other’s being a sum of non-negative numbers).
(b) Apply (a) for M1 ; M2 X M1 ; M3 X M2 ; : : : .
S
(c) Set Nn D M1 X Mn . Then M D M1 X Nn . Use (b). t
u
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4 The integral over a set 125
4.3.1 Remark R
For the general statement, the assumption
R that
R M f make sense is essential. The
point is that we could have both M f C and M f infinite.
For many purposes, we need a criterion by which sets and functions are measurable.
Let us begin with the following definition: For a Borel set X Rm , a function
f W X ! h1; 1i is called Borel measurable if
Comment: Note that since the inverse image preserves unions, intersections and
complements, we may equivalently replace every Borel set S in (C) by either every
interval h1; a/, a 2 R or every interval .a; 1i, a 2 R.
Proof of the Theorem: We begin by considering the easy implication. First,
suppose f is Borel measurable. Then so are f C and f , so by Proposition 2.5,
we may assume f 0. Then define
k k kC1
fn .x/ D n
when n f .x/ < : (*)
2 2 2n
Then clearly
fn % f:
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126 5 Integration II: Measurable Functions, Measure and the Techniques: : :
R
4.4.3 Lemma. If f 0, f is Borel measurable, and f D 0, then f D 0 almost
everywhere.
A contradiction. t
u
Proof. The statement for f is immediate. For f C g, note that .f C g/.x/ < a
if and only if there exist rational numbers q, r such that f .x/ < q, g.x/ < r and
q C r < a and thus, .f C g/1 Œ.1; a/ is the (countable) union of the Borel sets
f 1 Œ.1; q/ \ g 1 Œ.1; r/. t
u
Now let f be measurable. Then by Lemma 4.4 of Chapter 4, and Proposition 2.5,
it suffices to prove the statement for f C , f , and hence, by Lemma 4.4.2, for
f 2 L.
When f 2 L, by 4.7.5, there exist gn 2 Zdn such that
gn gnC1 f
and
Z Z
gn % f:
hn hnC1 f
and
Z Z
hn & f:
g D lim gn ; h D lim hn ;
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5 Parameters 127
gf h
and
Z
.h g/ D 0:
5 Parameters
Proof. Choose tn 2 U X ft0 g such that lim tn D t0 and use the Lebesgue Dominated
n
Convergence Theorem. t
u
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128 5 Integration II: Measurable Functions, Measure and the Techniques: : :
R
.3/ and for t 2 U there existZ f .t; /.
@f .t0 ; /
Then there exist the integral and one has
@t
Z Z
@f .t0 ; / d
D f .t0 ; /:
@t dt
@f .t0 ; x/ 1
Proof. We have D lim .f .t0 C h; x/ f .t0 ; x//. Set '.h; x/ D
@t h!0 h
1
h .f .t0 C h; x/ f .t0 ; x//. By Lagrange’s Theorem we have
ˇ ˇ
ˇ @f .t0 C h; x/ ˇ
j'.h; x/j D ˇˇ ˇ g.x/
ˇ
@t
6 Fubini’s Theorem
In this section we will have to indicate the dimension of the Euclidean space
we work in. When working in Rm , we will decorate the symbols Z; Zup ; L
up
etc. with subscripts Zm ; Zm ; Lm etc., and for the integral symbols we will use
R .m/ R .m/ R .m/ R R R
; ; instead of ; ; .
We will abandon
R the integral symbol I since we already know that for f 2 Z
we have If D f .
Finally, to avoid confusion in the case of two variables we will sometimes use
the classical
Z Z Z Z
f .x; y/dy or f .x; y/dx for f .x; / or f .; y/:
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6 Fubini’s Theorem 129
Z .mCn/ Z .mCn/
F .x/ D f .x; y/dy (resp. F .x/ D f .x; y/dy /:
F 2 Zm :
and F is continuous.
II. Now let fk 2 ZmCn , fk %k f . Then
Z
Fk .x/ D fk .x; y/dy % F .x/ and also fk .x; / % f .x; /
III. Now let f be general and let g 2 Zup be such that g f . Put G.x/ D
R .mCn/
g.x; y/dy, Then G F , and by II we have
gD G F
and hence
Z Z Z
f D inff g j g 2 Zup ; g f g F: t
u
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130 5 Integration II: Measurable Functions, Measure and the Techniques: : :
Theorem. (Fubini) Let f 2 LmCn . Then for almost all x there exists the integral
R .mCn/
f .x; y/dy. If we denote its value by F .x/, and define the values F .x/
arbitrarily in the remaining points, we have F 2 Lm and
Z .mCn/ Z .m/
f D F:
R R
Proof. Put F .x/ D f .x; y/dy and F .x/ D f .x; y/dy. By Lemma 6.1, we
have
8 Z 9
ˆ
ˆ >
>
Z Z Z ˆ
< F >
= Z Z Z
f D f F Z F f D f:
ˆ
ˆ >
>
:̂ F >
;
R
R f be in LmCn . Then the values are finite and we obtain, first ofR all, thatR F D
Let
F is finite and hence F 2 Lm , and similarly F 2 Lm . Further, F D F and
R
hence .F F / D 0 and hence F F D 0 almost everywhere, by 4.7. If f 2 LmCn
use Lebesgue’s Monotone Convergence Theorem. t
u
In this section, we will prove a substitution theorem for multivariable integrals. The
reader should be aware that a much more general substitution theorem is valid (see
[18]). In this text, we would basically be happy with a substitution theorem for the
Riemann integral of a continuous bounded function where the coordinate change is
a diffeomorphism with bounded partial derivatives (as needed, for example, in the
Stokes Theorem in Chapter 12 below). However, we will typically need to integrate
over Borel sets, which makes Lebesgue integral relevant. The purpose of this section
is to give a rigorous, but otherwise as straightforward as possible, proof of the
version of the theorem needed here.
7.1
Recall the set B of all Borel sets in Rm . Let U Rm be an open set. Define
BU D fS 2 BjS U g:
Note that clearly, BU is the smallest set of subsets of U closed under complements
and countable unions, which contains all open subsets of U . Let us also write
IU D fha1 ; b1 /
han ; bn /jha1 ; b1 i
han ; bn i U g:
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7 The Substitution Theorem 131
Proof. Let
S0 D ha1 ; b1 /
han ; bn /:
hr1 ; s1 /
hrn ; sn / .U X .T0 [ [ Tk1 //
where
si D ri C di .k/; ri D ai C `i di .k/
fS1 ; S2 ; : : : g D T1 [ T2 [ : : : :
S
By definition, the Si ’s are disjoint and one easily checks that Si is open and
closed in U . t
u
1 1
Vk D .a1 ; b1 /
.an ; bn /: t
u
k k
7.4 Proposition. Let SU be the smallest set of subsets of U which satisfies
.1/ Iu SU ;
.2/ When S1 ; S2 ; 2 SU are disjoint, then
1
[
Si 2 SU ; (C)
i D1
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132 5 Integration II: Measurable Functions, Measure and the Techniques: : :
SU .S / D fT 2 SU j S \ T 2 SU g: (7.4.1)
Step 1: If S 2 SU , then the conditions (2) and (3) above hold with SU replaced
by SU .S /.
S \ .U X T / D U X ..U X S / [ .S \ T // 2 SU ;
since .U X T / \ .S \ T / D ;. t
u
7.5 Assumption
F W U ! Rm
(Then F is regular, and by 7.2, 7.3 of Chapter 3, its image is open and its inverse
also satisfies the Assumption). Recall 3.2 of Chapter 3 for a discussion of DFx . The
attentive reader has noticed that
det.DFx /
is a special case of the Jacobian considered in 6.2 of Chapter 3 when the variables x
of 6.2 of Chapter 3 are not present and y is labeled as x. Many texts, in fact, reserve
the term for this special case.
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7 The Substitution Theorem 133
Proof. Note first that by Lemma 7.3 and the fact that F is a homeomorphism onto
its image, FŒS is Borel.
Next, one proves (*) in the case when is an affine map (see 5.9 of Appendix A).
By the multiplicative property of the determinant with respect to composition,
translation-invariance of Lebesge measure, Fubini’s Theorem and Gauss elimina-
tion, it then suffices to prove (*) for n D 1 (which is obvious) and for the map
1a
: (C)
01
For the case of (C), since is clearly invariant under translation, it suffices to prove
the statement for
S D h0; b1 /
h0; b2 /; b1 ; b2 > 0:
Then
[
n1
iab2 b2 iab2 b2 i b2 .i C 1/b2
FŒS h jaj ; C b1 C jaj /
h ; /:
i D0
n n n n n n
we have
ˇ ˇ
ˇ @Fi .y/ @Fi .a/ ˇ
ˇ ˇ
ˇ @x @x ˇ < ":
j j
FŒha1 ; b1 /
han ; bn /
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134 5 Integration II: Measurable Functions, Measure and the Techniques: : :
is a subset of
.FŒha1 ; b1 /
han ; bn // .1 C 2"/mjdet.DFx /j:
Proof. Let
S D ha1 ; b1 /
han ; bn /:
Sk .i1 ; : : : ; in /
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8 Hölder’s inequality, Minkowski’s inequality and Lp -spaces 135
Z Z
f D .f ı F/jdet.DFx /jdx:
FŒV V
Proof. First note that the statement for S 2 IU follows from the statement for V
open by Lemma 7.3. For V open, the inequality follows from Lemma 7.7. The
inequality follows from Lemma 7.7 with f replaced by f ı F, F replaced by F1 ,
FŒU replaced by U and V replaced by FŒV (recall that the set FŒU is open). u
t
7.9 Theorem. (The Substitution Theorem) Let F satisfy Assumption 7.5, and let
f W FŒU ! R be a continuous function. Let S 2 BU . Then
Z Z
f D .f ı F/jdet.DFx /jdx; (C)
FŒS S
provided that the integral on at least one side of the equation exists and is finite.
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136 5 Integration II: Measurable Functions, Measure and the Techniques: : :
1 1
8.1 Theorem. (Hölder’s inequality) Let p; q > 1 and let C D 1. We have
p q
Z
jfgj kf kp kgkq :
jf .x/j jg.x/j
jf .x/g.x/j C ;
p q
and hence
Z Z Z Z
11 1 1 1 1
jfgj D jf gj jf jp C jgjq D C D 1;
˛ˇ p q p q
and finally
Z
jfgj ˛ˇ D kf kp kgkq : t
u
Remark: The equality holds if and only if the functions are dependent, but we
will not need the other implication.
and hence
1 p 1 pCq 1
kf kp kgkq D ˛ q .kf kp /1C q D ˛ q .kf kpp / pq D ˛ q kf kpp :
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8 Hölder’s inequality, Minkowski’s inequality and Lp -spaces 137
kf C gkp kf kp C kgkp
Proof. The inequality is obvious for p D 1 and p D 1, hence we can assume that
1 > p > 1.
Recall Proposition 4.5.2 of Chapter 1. For p 1 and x 0, the function f .x/ D
x p is convex (since h00 .x/ D p.p 1/x p2 0) and hence we have
1 1 1 1
jf C gjp . j2f j C j2gj/p D j2f jp C j2gjp D 2p1 jf jp C 2p1 jgjp :
2 2 2 2
R R
RThus, first,pif the integrals jf j and jgj are finite, also the integral of the sum
p p
1 1 p1
Proceed, using Hölder inequality, taking into account that D 1 D and
q p p
p1
hence q D ,
p
Z Z Z p
1 1
.p1/ p1 1 p1
.. jf jp / p C. jgjp / p /. jf Cgj / D .kf kp Ckgkp /.kf Cgkp /p1 :
Hence
and Minkowski’s inequality follows dividing both sides by .kf C gkp /p1 . t
u
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138 5 Integration II: Measurable Functions, Measure and the Techniques: : :
By Theorem 8.2, Lp .B/ is a vector space over R, and it may appear that
therefore defines a norm on Lp .B/ in the sense of 1.2.1 of Chapter 2. This is,
however, not true for the simple reason that two functions f; g which are equal
almost everywhere have 0 distance! It is immediately obvious, on the other hand,
that the converse is also true, since we have the following fact.
R
8.3.1 Lemma. If f W B ! Œ0; 1 and X f D 0, then f D 0 almost everywhere
on B.
R
Proof. Let, for " > 0, E" D fx 2 X jf .x/ > "g. Then clearly X f > ".E" /,
so .E" / D 0. The set E D E1=1 [ E1=2 [ [ E1=n [ : : : therefore satisfies
.E/ D 0, but we have E D fx 2 X jf .x/ ¤ 0g. t
u
Thus, we see that (8.3.1) gives a well-defined norm on the quotient space
Lp .B/ D Lp .B/=L0
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8 Hölder’s inequality, Minkowski’s inequality and Lp -spaces 139
jjf C gjjp jj jf j C jgj jjp jj jf j jjp C jj jgj jjp D jjf jjp C jjgjjp :
The case of p D 1 needs a separate (easy) discussion, see Exercise (17). Note that
a complex analogue of Hölder’s inequality follows from the real case immediately.
Thus, we can define the normed vector spaces Lp .B; C/, 1 p 1 completely
analogously as the spaces Lp .B/, with real functions replaced by complex ones.
while gn % lim inf fn , so the statement follows by passing to the limit by the
n!1
Lebesgue Monotone Convergence Theorem. t
u
8.5.2 Theorem. The spaces Lp .B/ and Lp .B; C/, 1 p 1, are complete
metric spaces.
Proof. Consider, for example, the complex case (the proof in the real case is the
same). Let fn W X ! C represent a Cauchy sequence in Lp . Then there exist
n1 < n2 < < nk < : : : such that
1
X
jjfnk fnkC1 jjp < 1:
kD1
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140 5 Integration II: Measurable Functions, Measure and the Techniques: : :
in that case, the convergence is uniform (Exercise (18)). Now let f .x/ D lim fnk .x/
for x 2 S , and f .x/ D 0 for x 2 X X S . In the case of p D 1, we are done. For
p < 1, by Fatou’s Lemma 8.5.1,
Z Z
jfn f jp lim inf jfn fnk jp : (8.5.1)
B k!1 B
If we choose n such that jjfn fm jjp < ", then the right-hand side of (8.5.1) is ".
The right-hand side of (8.5.1) converges to 0 with n ! 1 because the sequence fn
is Cauchy. t
u
8.6.1 Lemma. Let 1 < p and let B Rn be a Borel subset such that .B/ < 1.
Then
Z p Z
1 1
jf .x/j jf .x/jp :
.B/ B .B/ B
Proof. Put
Z
1
x0 D jf .x/j:
.B/ B
Since .x p /00 > 0 on .0; 1/, the derivative of x p is increasing on .0; 1/. Therefore,
if we let b D .x0 /p and let a be the value of .x p /0 D px p1 at x0 , we have
ax0 C b D .x0 /p
Now compute:
Z Z
1 1
jf .x/j
p
.ajf .x/j C b/ D ax0 C b D .x0 /p ;
.B/ n .B/ B
as claimed. t
u
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9 Exercises 141
9 Exercises
T
(b) Now let A1 A2 : : : , A D Ai . Give an example when (*) does not
hold. Formulate a reasonable hypothesis which fixes the problem. [Hint:
Finiteness.]
(6) Let M Rm be a measurable set, and let f W M ! R be a function such
that for every Borel set S Rm , f 1 ŒS is measurable. Prove that then the
functionf defined by
(
f .x/ for x 2 M;
f .x/ D
0 otherwise
is measurable.
(7) Give an example of a measurable function f W Rm ! R such that there exists
a measurable set S R where f 1 ŒS is not measurable.
(8) Prove the following strengthening of Corollary 4.3.1: Let S be a Lebesgue
measurable set in Rm . Then there exists a subset K S of type F (a
countable union of compact sets) such that .S X K/ D 0. [Hint: First note
that for a real function f 2 Zdn , f 1 Œha; 1/ is closed. Now in the proof of
4.4, we produced a non-decreasing sequence of Zdn -functions fn cS such
that fn % cS almost everywhere. Let K be the union of fn1 Œh1=2; 1/.]
(9) Prove that if S is a Lebesgue measurable set in Rm , then there exists a set
U of type Gı (countable intersection of open sets) containing S such that
.U X S / D 0.
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142 5 Integration II: Measurable Functions, Measure and the Techniques: : :
for 0 < a < . [Hint: Find the derivative with respect to a first.]
(12) Compute
Z
xy
E
m W .0; 1/
. =2; =2/n2
.0; 2 / ! Rm
x1 D r cos.t1 / : : : cos.tm1 /;
xi D r cos.t1 / : : : cos.tmi / sin.tmi C1 / i D 2; : : : m:
Prove that
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9 Exercises 143
and is given by
R2
(and similarly for Cn ). Prove that this makes Rn , Cn into normed vector spaces.
What is the appropriate definition in the case of p D 1?
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1 The problem
1.1
where fk are continuous functions of n C 1 real variables. Note that then yi , since
they are required to have a derivative, must in particular be continuous, and the
derivative is then also continuous by (1.1.1). The expression “ordinary” indicates
that there appear only derivatives of functions of one variable, not partial derivatives
of functions of several variables.
Using the vector symbols y, f as in Chapter 3, we can describe the task by writing
1.2
This appears to call for a generalization of the original problem. But in fact, such
systems are easily converted to systems of ODE’s as above: in this particular case,
introduce additional variables
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146 6 Systems of Ordinary Differential Equations
making the two equations into the equivalent system of the form (1.1.1):
z01 D z3 ;
z02 D z4 ;
z03 D z5 ;
z04 D z6 ;
z05 D z7 ;
z06 D f2 .x; z1 ; : : : ; z7 /;
z07 D f1 .x; z1 ; : : : ; z6 ; f2 .x; z1 ; : : : ; z7 //:
The reader certainly sees how to apply this procedure in a general situation
.k / .k / .k /
y1 1 D f1 .x; y1 ; : : : ; y1 1 ; : : : ; yn ; : : : ; yn n /;
::: (1.2.1)
.k / .k / .k /
yn n D fn .x; y1 ; : : : ; y1 1 ; : : : ; yn ; : : : ; yn n /:
Introduce additional variables for all the derivatives of yi of order less than the
highest order derivative of yi which occurs in the system, and rewrite the original
system in terms of the additional variables, introducing additional equations relating
the new variables as derivatives of each other (see Exercise (1), (2)). To be explicit,
one sometimes refers to a system of the form (1.1.1) as a system of first-order ODE’s,
but we already see that such systems are all we need to consider.
1.3
We may, in fact, encounter even more general systems, namely a system of equations
of the form
.k / .k /
F1 .x; y1 ; : : : ; y1 1 ; : : : ; yn ; : : : ; yn n / D 0;
::: (1.3.1)
.k / .k /
Fm .x; y1 ; : : : ; y1 1 ; : : : ; yn ; : : : ; yn n / D 0:
In such a case, we will always assume that m D n and that the Jacobian of the
.k / .k /
Fi ’s in the variables corresponding to y1 1 ; : : : ; yn n is non-zero. Then, using the
Implicit Function Theorem 6.3 of Chapter 3, the system (1.3.1) can be converted (at
least locally) to the system (1.2.1), and hence again, by the method explained there,
to a first-order system of the form (1.1.1). If m ¤ n or the Jacobian in question is 0,
the problem (1.3.1) will be considered ill-posed from our point of view.
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2 Converting a system of ODE’s to a system of integral equations 147
Note that whether the problem (1.3.1) is well-posed depends on the values of x,
.k 1/
the yi ’s and their derivatives up to yi i , and the (number) solution of the resulting
.k /
equations for the yi i ’s. We will see, however, that this is in the spirit of the theory
we will develop, as in solving the system (1.2.1), we get to specify x, the yi ’s and
.k 1/
their derivatives up to yi i as initial conditions. (This is equivalent to specifying
x and yi as initial condition in the system 1.1.)
The translations of 1.2 and 1.3 serve a theoretical purpose. They may often be
difficult to carry out in practice. In many cases, different reductions may be more
advantageous. (See Exercise (3).)
1.4 Remarks
in this interval such that, moreover, yj .x0 / D j if and only if they satisfy
the equations
Z x
yj .x/ D fj .t; y1 .t/; : : : ; yn .t//dt C j : (2.1.2)
x0
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148 6 Systems of Ordinary Differential Equations
On the other hand, if the functions yj .x/ satisfy (2.1.2) then by taking the derivative
by x we obtain that yj0 .x/ D fj .x; y1 .x/; : : : ; yn .x//, and setting x D x0 we
conclude that yj .x0 / D j . t
u
2.2 Remark
This very easy translation of our problem has in fact a quite surprising consequence.
Let us illustrate it on the equation y 0 D f .x; y/. Denote by D the operator of taking
the derivative, and by F the operator transforming y.x/ to f .x; y.x//. Further,
define an operator J by setting
Z x
J.y/.x/ D f .t; y.t//dt:
co
This looks somewhat scary: for example, if we take the space X D C..a; b//
of bounded continuous functions on .a; b/ as considered in 7.7 of Chapter 2,
the operator D is not even defined on X , as not every continuous function has
a derivative. It seems that in order to treat the equation by means of spaces of
functions, we would have to think hard what space to work on, and what metric
to choose to make both sides of the equation (*) continuous. Such problems do,
indeed, arise with some types of differential equations.
However, in case of our system (1.1.1), Theorem 2.1 gives a way out: After the
translation we obtain the equation
y D J.y/ (**)
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3 The Lipschitz property and a solution of the integral equation 149
3.1
U D .x0 ˛ 0 ; x0 C ˛ 0 /
.y1 ˇ 0 ; y1 C ˇ 0 /
.yn ˇ 0 ; yn C ˇ 0 /
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150 6 Systems of Ordinary Differential Equations
on which f jU is Lipschitz. Now choose 0 < ˛ < ˛ 0 and 0 < ˇ < ˇ 0 . We have an
M such that
implies that
jfj .x; y1 ; : : : ; yn /j A
C D C..x0 a; x0 C a//
Yj D fu j u 2 C; j ˇ u.x/ j C ˇg:
All the Yj are complete metric spaces and hence also the product
Y D Y1
Y2
Yn
where j .; / D supx j.x/ .x/j, is complete (7.7.2 and 7.3.1 of Chapter 2).
Now define for u D .u1 ; : : : ; un /
where
Z x
Jj .u/.x/ D fj .t; u1 .t/; : : : ; un .t//dt C j :
x0
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4 Existence and uniqueness of a solution of an ODE system 151
Since
ˇZ x ˇ
ˇ ˇ
jJj .u/.x/ j j D ˇˇ fj .t; u1 .t/; : : : ; un .t//dt ˇˇ
x0
Z x
jfj .t; u1 .t/; : : : ; un .t//jdt jx0 xj A a A ˇ;
x0
Since we have jfk .t; u1 .t/; : : : / fk .t; v1 .t/; : : : /j M max xjuj .t/ vj .t/j
j
M max sup juj .x/ vj .x/j D M .u; v/ we obtain
j x
Thus, J W Y ! Y satisfies the condition of the Banach Fixed Point Theorem 7.6 of
Chapter 2 and we conclude that there is precisely one u such that J.u/ D u, that is,
precisely one solution of our integral equations on the interval .x0 a; x0 C a/. u
t
4.1
has precisely one solution on .x0 a; x0 C a/ such that yj .x0 / D j for all j .
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152 6 Systems of Ordinary Differential Equations
Remark. Thus, unlike the uniqueness in 3.3, the solution is unique with respect
to the extra conditions yj .x0 / D j . These requirements are usually referred to as
the initial conditions.
4.2
The solutions in 4.1 are of a local character, that is, they are guaranteed in a small
neighborhood of the initial point x0 only. Now we will head to solutions of a more
global character, defined as far as possible. To start with, we will speak of a local
solution .u; J / defined on an open interval J and we will endeavour to extend the J .
4.2.1 Lemma. Under the conditions of 4.1, let J; K be open intervals, let x0 2
J \ K, and let .u; J / and .v; K/ be local solutions such that u.x0 / D v.x0 /. If f is
continuous and Lipschitz with respect to the yj in the domain in which we consider
our system, we have ujJ \ K D vjJ \ K.
Proof. By 4.1, if the u and v coincide at a point they coincide in some of its open
neighborhoods. Thus,
U D fx j u.x/ D v.x/; x 2 J \ Kg
4.2.2
Take the union of all the intervals J on which there exists a solution u satisfying
uj .x0 / D j . By Lemma 4.2.1, there exists a solution .u; J / with the domain J .
Such maximal solutions are called the characteristics of the given ODE system. In
this terminology we can summarize the preceding facts in the following
4.3
From the method of 1.2 and from Theorem 4.2.2, we obtain the following
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5 Stability of solutions 153
y k .x0 / D kC1 ; k D 0; : : : ; n 1:
4.4 Examples
y0 D 1 C y2
y.x/ D 0 for a x b;
ˆ
:̂.x b/3 for x b;
5 Stability of solutions
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154 6 Systems of Ordinary Differential Equations
using of the solution in practical applications would be rather suspect, as the effect
of small errors in initial conditions would be unpredictable.
Furthermore, a practical setting often contains additional parameters, so the
system becomes
and define
gj .x; z1 ; : : : ; zn ; ˛1 ; : : : ; ˛k ; ˇ1 ; : : : ; ˇn / D fj .x; z1 C ˇ1 ; : : : ; zn C ˇn ; ˛1 ; : : : ; ˛k /
z0j .x;˛1 ; : : : ; ˛k ; ˇ1 ; : : : ; ˇn /
D gj .x; z1 .x; ˛1 ; : : : ; ˛k ; ˇ1 ; : : : ; ˇn /; : : : ; zn .x; ˛1 ; : : : ; ˛k ; ˇ1 ; : : : ; ˇn //;
zj .x0 ;˛1 ; : : : ; ˛k ; ˇ1 ; : : : ; ˇn / D 0j ; j D 1; : : : ; n
with the initial values 0j fixed. Thus, it suffices to study the dependence of the
system on parameters only, with initial conditions fixed; in the notation (*), this
means we will study stability with respect to ˛1 ; : : : ; ˛k , with j fixed.
5.1.1 Remark
One can also convert the combined stability problem into a problem concerning
initial conditions only. But the trick with parameters is more expedient and we will
concentrate on that.
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5 Stability of solutions 155
F .x/ C eK.xa/ :
Proof. Put
Z x
G.x/ D C C K F .t/dt:
a
Then we have
so that
5.3
To simplify notation, in the proof of the following theorem we will write ˛ for
˛1 ; : : : ; ˛k and use the symbol
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156 6 Systems of Ordinary Differential Equations
for all j D 1; : : : ; n.
Proof. We have
so
Z x
ky.x; ˛/ y.x; ˇ/k K .k˛ ˇk C ky.t; ˛/ y.t; ˇ/k/dt:
a
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5 Stability of solutions 157
F .x/ k˛ ˇkeK.xa/
and since ky.x; ˛/ y.x; ˇ/k F .x/, the estimate (5.3.2) follows. t
u
5.4 Remark
Recall that the existence and uniqueness in Theorem 4.1 was proved using the
Banach Fixed Point Theorem 7.6 of Chapter 2. The reader may naturally ask
whether the stability theorem (at least the continuity) is not an easy consequence of
a general property of such fixed points. That is, we think of the following problem.
Let us have metric spaces X; T and a mapping
f WX
T !X
such that d.f .x; t/; f .y; t// rt where rt < 1 depend on t 2 T only. Define
F .t/ 2 X by the equation f .F .t/; t/ D F .t/. How does F .t/ depend on t?
There are fairly general facts known on this subject, but they do not fit well with
our present topic. Due to the special character of our equations it is, luckily enough,
easy to show the dependence by an explicit estimate, as we have done.
5.5
@yi
satisfying the conditions from 4.1 (where we write, similarly as before, yi0 , not ,
@x
for the derivatives by x, to keep in mind the fact that we are dealing with an ordinary
differential equation).
@yi
zi .x; ˛/ D .x; ˛/
@˛p
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158 6 Systems of Ordinary Differential Equations
zi .x0 ; ˛/ D 0; (5.5.2)
Remarks.
1. The continuous differentiability with respect to yj and ˛p makes, of course, the
functions fi locally Lipschitz with respect to these variables.
2. The system (5.5.1) is viewed as solved and yi .x; ˛/ constitute the (unique)
solution. The equations (5.5.2) contains these functions as aleady given, not as
something dependent on the zi . Thus, the right-hand sides of the equations in
(5.5.2) are Lipschitz with respect to zj and therefore the system has a solution.
Our task will be to prove that the individual zi ’s are the partial derivatives of the
yi by ˛p .
3. The reader has certainly not overlooked that the equations for zi which we hoped
@yi
to be the come naturally in the form (5.5.2): if we already knew yi to have
@˛
derivatives, we would obtain the equality by taking derivatives of the equalities
in (5.5.1). But this we do not know yet.
Proof. First of all, note that the problem is immediately reduced to the case k D 1:
We may treat all parameters but one as constant for the existence of a single
partial derivative; once equation (5.5.2) is proved, we can use Theorem 5.3 to prove
continuity of the partial derivatives in all the ˛p ’s. Thus, let us assume k D 1, and
write ˛ for ˛p .
Let yi be a solution of the system (5.5.1) and z a solution of the system (5.5.2).
Put
1
ui .x; ˛; h/ D .yi .x; ˛ C h/ yi .x; ˛//
h
and
Thus,
@vi @ui
.x; ˛; h/ D .x; ˛; h/ z0i .x; ˛/
@x @x
@ui X n
@fi @fi
D .x; ˛; h/ .x; y.x; ˛/; ˛/ zj .x; y.x; ˛/; ˛/:
@x j D1
@yj @˛
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5 Stability of solutions 159
@ui
Let us compute the derivative .x; ˛; h/ :
@x
@ui 1
.x; ˛; h/ D .yi0 .x; ˛ C h/ yi0 .x; ˛//
@x h
1 X
n
D . fj .x; y.x; ˛ C h/; ˛ C h/ f .x; y.x; ˛/; ˛ C h//
h j D1
1
C .fi .x; y.x; ˛/; ˛ C h/ f .x; y.x; ˛/; ˛//:
h
X n
@fi @fi
D .x; y.x; ˛/ C 1
y; ˛ C h/ uj .x; ˛; h/ C .x; y.x; ˛/; ˛ C 2 h/:
j D1
@yj @˛
@vi
Let us now consider . Since ui .x; ˛; h/ D vi .x; ˛; h/ C zi .x; ˛/, we obtain
@x
ˇ ˇ Xn ˇ ˇ
ˇ @vi ˇ ˇ @fj ˇ
ˇ .x; ˛; h/ ˇ ˇ .x; y.x; ˛/ C
y; ˛ C h/ ˇ jvj .x; ˛; h/j
ˇ @x ˇ ˇ @y 1 ˇ
j D1
n ˇ
X ˇ
ˇ @fi @fj ˇ
C ˇ. .x; y.x; ˛/ C
y; ˛ C h/ .x; y.x; ˛/; ˛// z .x; ˛/ ˇ
ˇ @y 1
@y
j ˇ
j
j D1
ˇ ˇ
ˇ @fi @fi ˇ
ˇ
Cˇ .x; y.x; ˛/ C 2 h/ .x; y.x; ˛//ˇˇ ;
@˛ @˛
and further
ˇ ˇ Xn ˇ ˇ
ˇ @vi ˇ ˇ @fi ˇ
ˇ .x; ˛; h/ ˇ ˇ .x; y.x; ˛/ C
y; ˛ C h/ ˇ jvj .x; ˛; h/j
ˇ @x ˇ ˇ @y 1 ˇ
j D1 j
n ˇ
X ˇ
ˇ @fi @fj ˇ
C ˇ. .x; y.x; ˛/C
y; ˛ C h/ .x; y.x; ˛/; ˛ C h// z .x; ˛/ ˇ
ˇ @y 1
@y
j ˇ
j D1 j
n ˇ
X ˇ
ˇ @fi @fj ˇ
C ˇ. .x; y.x; ˛/; ˛ C h/ .x; y.x; ˛/; ˛// z .x; ˛/ ˇ
ˇ @y @y
j ˇ
j
j D1
ˇ ˇ
ˇ @fi @fi ˇ
C ˇˇ .x; y.x; ˛/ C 2 h/ .x; y.x; ˛//ˇˇ :
@˛ @˛
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160 6 Systems of Ordinary Differential Equations
Now let " > 0. From the Lipschitz property we see that for h sufficiently small we
have for all x sufficiently close to x0 to stay in the aforementioned range
X n ˇ ˇ
ˇ @fi @fj ˇ
ˇ. ˇ
ˇ @y .x; y.x; ˛/ C 1
y; ˛ C h/ @y .x; y.x; ˛/; ˛ C h// zj .x; ˛/ˇ
j D1 j
X n ˇ ˇ
ˇ @fi @fj ˇ
C ˇ ˇ
ˇ. @y .x; y.x; ˛/; ˛ C h/ @y .x; y.x; ˛/; ˛// zj .x; ˛/ˇ
j D1 j
ˇ ˇ
ˇ @f @f ˇ
ˇ
C ˇ .x; y.x; ˛/ C 2 h/ .x; y.x; ˛//ˇˇ < "
@˛ @˛
and hence
ˇ ˇ X
ˇ @vi ˇ n
ˇ .x; ˛; h/ ˇ "CK jvj .x; ˛; h/j
ˇ @x ˇ
j D1
so that
n ˇ
X
ˇ
X
ˇ @vi ˇ n
ˇ .x; ˛; h/ ˇ "CK jvj .x; ˛; h/j;
ˇ @x ˇ
i D1 j D1
and consequently
X
n Z x X
n
jvi .x; ˛; h/j .n" C nK jvi .t; ˛; h/j/dt:
j D1 x0 i D1
X
n
Thus, for F .x/ D n" C nK lim jv.x; ˛; h/j we have
i D1
Z x
F .x/ F .t/dt
x0
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6 A few special differential equations 161
1
and since " > 0 was arbitrary we conclude that lim .yi .x; ˛ C h/ yi .x; ˛// D
h!0 h
zi .x; ˛/: t
u
6.1
First of all, let us realize that in the situations where the theorem on the existence and
uniqueness is applicable, we do not really have to be concerned about the correctness
dy
of the procedure we use (e.g. working with as if it were a fraction, failing to
dx
control whether there might not be a zero in a denominator, etc.). If we obtain a
function satisfying the equation (and initial conditions), it has to be the one and
only solution we are looking for, by Theorem 4.2.2. This is a perfect example of the
importance of theoretical work for calculations.
6.2
y 0 D f .x/:
The equation
y 0 D f .x/g.y/
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162 6 Systems of Ordinary Differential Equations
Z Z
1
. /.y.x// D . f /.x/ C C:
g
This somewhat clumsy computation can be, more intuitively, modified as follows.
Take the equation as
dy
D f .x/g.y/;
dx
proceed to
dy
D f .x/dx
g.y/
and “integrate”
Z Z
dy
D f .x/dx C C:
g.y/
Examples.
1. For y 0 D y sin x we obtain
Z Z
dy
D sin xdx C C;
y
hence
ln y D cos x C C
yielding
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6 A few special differential equations 163
6.4
y 0 D f .ax C by/;
6.5
(in other words, y 0 D F .x; y/ where F is such that for any t, F .x; y/ D F .tx; ty/),
y
substitute z D . Then we obtain
x
y0x y y0 z 1
z0 D 2
D D .f .z/ z/ ;
x x x
again an equation with separated variables.
6.6
The equation
ax C by C c
y0 D f (6.6.1)
˛x C ˇy C
would be of the type 6.5 if we had c D D 0. If not, let us try to force it. Let x0 ; y0
be a solution of the linear (algebraic) equations
ax C by C c D 0
˛x C ˇy C D 0:
Then
ax C by C c a.x x0 / C b.y y0 /
D :
˛x C ˇy C ˛.x x0 / C ˇ.y y0 /
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164 6 Systems of Ordinary Differential Equations
If we substitute
D x x0 ; z D y y0 ;
The linear algebraic equations above may fail to have a solution: namely we could
have had .a; b/ D K .˛; ˇ/ or K .a; b/ D .˛; ˇ/. Then, however, the equation 6.6.1
is already of the form y 0 D F .Ax C By/ as it is, and we can use the procedure
from 6.4.
First, solve the equation y 0 D a.x/y. This is a case of separated variables and by
the method from 6.3, we obtain a solution
R
u1 .x/ D c e a.x/dx
: (6.7.1)
y 0 D c 0 u1 C cu01
y 0 D c 0 u1 C cau1 D c 0 u1 C ay:
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7 General substitution, symmetry and infinitesimal symmetry of a differential equation 165
y 00 D f .y/:
Such an equation can be solved as follows. First, multiply both sides by y 0 to obtain
y 0 y 00 D f .y/y 0 ;
that is,
Z Z
1 1 0 2
. .y 0 /2 /0 D .. f / ı y/0 and further .y / D . f/ıy CC
2 2
7.1
One may ask how, looking at a differential equation, one finds the substitution which
allows us to separate variables. Of course, in most cases, it is not possible. When it
is, however, there is, in fact, a general strategy for finding the substitution, relating
separation of variables to symmetry. To study symmetry, it is convenient to write
a system of differential equations in a form in which the right-hand side does not
depend explicitly on x:
Clearly, this is a special case of the system (1.1.1). On the other hand, a system of the
form (1.1.1) can be always reduced to the form (7.1.1) by introducing an additional
variable y0 :
y00 D 1;
yi0 D fi .y0 ; y1 ; : : : ; yn /:
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166 6 Systems of Ordinary Differential Equations
7.2
Now assume we have a system of the form (7.1.1). We may write it in vector
notation, putting y D .y1 ; : : : ; yn /T , f D .f1 ; : : : ; fn /T (recall that reconciling the
direction of composition of maps with matrix multiplication favors viewing vectors
as columns here, see e.g. Appendix A, 7.5):
y0 D f.y/: (7.2.1)
Let us point out a geometric interpretation of the system (7.2.1). Denote the
independent variable by t. A solution y.t/ can be interpreted as a parametric curve
with the parameter t. Then the equation (7.2.1) says that the tangent (“velocity”)
vector of the curve y at the point t is equal to f.y.t//. A function U ! Rn on
a subset U yRn when we interpret its values as vectors is called a vector field.
The curves y.t/ are called integral curves of the vector field. One sometimes denotes
the solution as
although this is somewhat misleading, given the fact that the solution is not an
exponential even in the case of n D 1 unless f is constant, and cannot be figured out
explicitly in general when n > 1.
7.3
Let us now study how a vector field changes when we change variables. By a
substitution at y0 2 Rn we shall mean a smooth map W U ! Rn where U is an
open neighborhood of y0 whose differential at y0 is non-singular. Writing z D .y/,
then, by the chain rule, we get from (7.2.1) a system of differential equations for z,
z0 D Dj 1 .z/ f. 1 .z//;
(the operation on the right-hand side is matrix multiplication), so from the point of
view of differential equations, transforms the vector field f to the vector field
z0 D g.z/: (7.3.1)
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7 General substitution, symmetry and infinitesimal symmetry of a differential equation 167
7.4
We will call a symmetry (at y0 ) if the differential equations (7.2.1) and (7.3.1)
coincide, i.e. we have g.z/ D f.z/, or
0 D Id
(in other words, 0 .y/ D y). Given a family of symmetries, what is happening near
" D 0? Let
ˇ
@" ˇˇ
uD : (7.4.2)
@" ˇ"D0
Then considering the condition (7.4.1) for D " and differentiating by " at " D 0,
we get that
ˇ
@ ˇ
f." .y//ˇˇ D Df u.y/ D @u f .y/;
@" "D0
ˇ
@ ˇ
Dj.";y/ f.y/ˇˇ D Dujy f.y/ D @f u.y/
@" "D0
d
(here on the right-hand side we use the notation @u f.y/ D f .y C tu/, see 2.4 of
dt
Chapter 3).
7.5
and call this the Lie bracket of vector fields. This is, again, a vector field. The
derivative of the condition (7.4.1) at " D 0 then reads
Œu; f D 0: (7.5.1)
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168 6 Systems of Ordinary Differential Equations
7.6
It is worth pointing out two properties of the Lie bracket of vector fields:
Œu; Œv; w C Œv; Œw; u C Œw; Œu; v D 0: (7.6.2)
The equality (7.6.2) is called the Jacobi identity. Generally, a vector space over R
or C with a binary operation Œ‹; ‹ which is linear in each coordinate and satisfies
the equalities (7.6.1), (7.6.2) is called a Lie algebra. Thus, in particular, smooth
vector fields defined on the same open subset of Rn form a Lie algebra, as do
symmetries of the differential equation (7.2.1) at a given point y0 (this follows from
the Jacobi identity).
7.7 Comment
Several concepts of this and the next section are closely related to Chapter 12 below.
After finishing that chapter, the reader may be ready to tie this in together in some
highly interesting and important geometrical notions which are beyond the scope of
this text. For example, the notion of Lie algebra just mentioned leads to the notion
of a Lie group. In Chapter 12, we will develop enough techniques to introduce the
concept of a Lie group, and will mention it briefly in Exercises (6), (7), (8) of
Chapter 12. Lie groups are a major field of mathematical study. We recommend
[9, 10] for further reading.
8.1
exp."u/ (8.1.1)
(used in the sense of the notation (7.2.2)) is a continuous family of symmetries. This
is because in case of " equal to (8.1.1), by definition, the derivative of the condition
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8 Symmetry and separation of variables 169
(7.4.1) by " is the same at every point ", and is equal to Œu; f D 0. Now given an
infinitesimal symmetry u of the equation (7.2.1) at a point y0 , and assuming
u.y0 / ¤ 0; (8.1.2)
(By Steinitz’ Theorem 2.6 of Appendix A, this can be always achieved after
permuting the coordinates fi .) Assuming (8.1.3) holds, consider the following
smooth map U ! Rn defined in an open neighborhood U of y 0 :
ˆ.y0 / D y0 ;
(although obviously that is not important), and by (8.1.3) and the Implicit Function
Theorem, the map ˆ has a smooth inverse ‰ in an open neighborhood of y 0 . We
consider the substitution
z D ‰.y/: (8.1.5)
z1 7! z1 C "; (8.1.6)
zi 7! zi for i D 2; : : : ; n: (8.1.7)
This means that the function g does not depend on the variable z1 , and thus, we
have reduced the number of variables by 1: we have a system of n 1 differential
equations in the variables z2 ; : : : ; zn , and an equation for z01 in terms of z2 ; : : : ; zn .
For n D 2, this implies a complete solution (separation of variables). Of course,
to make this method work, we must be able to evaluate (8.1.1), which, a priori, is
a system of n differential equations. However, in some cases, symmetries may be
more easily visible than direct solutions.
8.2
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170 6 Systems of Ordinary Differential Equations
Œu; f D f (8.2.2)
g.z/=g1 .z/
dzi gi .z/
D ; i D 2; : : : n:
dz1 g1 .z/
Note, however, that now unless the factor ˛ of the generalized symmetry has some
special form, we still end up with a general first-order differential equation for the
variable z1 .
8.2.1 Example
Consider the homogeneous differential equation
y
y 0 D f . /:
x
In symmetric form, this is
y
y 0 D f . /;
x
x 0 D 1:
(to conform with the above notation, " D 1). The corresponding infinitesimal
symmetry is
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8 Symmetry and separation of variables 171
which exponentiates to
so (fixing, say, x 0 D 1 and calling the new variables z; v), the substitution becomes
or
z D 1 C ln.x/;
y (**)
vD :
y0x
Up to scalar multiple, the formula for v is the substitution from the last section. It
is worthwhile noting, however, that in the present form, we obtain the autonomous
equation
dv 1
D 0 .f .y 0 v/ v/
dz y
(which we may not have noticed in the last section). Obviously, the rather simple
form of the generalized infinitesimal symmetry allows us to recover z in this case.
8.3 Example
The fact that for n D 2, a symmetry leads to separation of variables, begs the
question whether the separated equation
y 0 D a.x/y (8.3.1)
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172 6 Systems of Ordinary Differential Equations
8.4 Example
The symmetry (8.3.2) (subject to the condition (8.3.3)) plainly also is a symmetry
of the equation
(since (8.3.2) has 0 Lie bracket with .0; b.x//T ). Thus, we may use this symmetry
to solve the equation (8.4.1). The substitution we get by choosing y 0 D .0; 0/,
y1 D y; y2 D x, is
y D z1 k.z2 /; x D z2 :
Setting z D z1 , we get
9 Exercises
z0 y 0
y 00 D C y3;
zCy Cx
z00 D ln.z0 C cos.y 0 C z// C 3
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9 Exercises 173
y 0 C sin.x C y C y 0 / D 0
2x
y0 D :
ey
(6) Solve the differential equation
x2 C y 2
y0 D :
xy
y 0 D f .ax C by/
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Systems of linear differential equations have many special properties, the most
important of which is that a characteristic is defined in any open interval in which
the system is defined (in contrast with ODE, see Example 4.4.1 of Chapter 6).
In this chapter, we prove this important “no blow-up” theorem, and discuss the
linear character of the set of solutions. We also describe a method for solving
completely the important class of systems of linear differential equations with
constant coefficients.
1.1
X
n
yi0 .x/ D aij .x/yj .x/ C bi .x/; i D 1; : : : ; n: (L)
j D1
Recall that such systems arise naturally as equations for partial derivatives of
solutions of general differential equations by a parameter (see (5.5.2) of Chapter 6).
A linear (differential) equation of order n, where ai ; b are continuous on J , is
Again, the system (L̃) is easy to translate to a system of the form (L) by the method
of 1.2. In fact, again, one may call (L) a system of first order LDE’s, define systems
of higher order LDE’s, and then show such systems are equivalent to systems of first
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176 7 Systems of Linear Differential Equations
order LDE using the method of 1.2 of Chapter 6. Consequently, it suffices, again, to
develop a theory for first-order systems (L). However, in some practical situations,
it is advantageous to treat the special case of a single higher order equation (L̃)
separately, as we will see below.
If all the functions bi are zero (in the case (L̃), if b is zero), we speak of
homogeneous equations resp. equation. The homogeneous counterpart of an (L)
resp. (L̃) will be indicated by (L-hom) resp. (L̃-hom).
1.2 Lemma. Let f be continuous and bounded on the half-open interval ha; b/.
Define a value of f at b arbitrarily. Then there exists the (Riemann) integral
Rb
a f .t/dt and we have
Z b Z x
f .t/dt D lim f .t/dt:
a x!b a
Comment: We prove this result here directly to make this chapter (and Chapter 6
above) largely self-contained, and independent of the techniques of the Lebesgue
integral as introduced in Chapters 4, 5. The attentive reader, however, should see
how the present statement follows from a much stronger result in Exercise (10) of
Chapter 5, Exercise (4) of Chapter 4, and the Lebesgue Dominated Convergence
Theorem.
Rx
Proof. The Riemann integrals a trivially exist (because of the continuity).
Let jf .x/j C . Thus, we can choose partitions D.x/ of ha; xi such that
Z x Z x
" "
f s.f jha; xi; D.x// S.f jha; xi; D.x// f C (*)
a 2 a 2
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1 The definition and the existence theorem for a system of linear differential equations 177
Z x Z b Z b Z x
f " f f f C ";
a a a a
hence
ˇZ Z b ˇˇ ˇZ Z b ˇˇ
ˇ x ˇ x
ˇ ˇ ˇ ˇ
ˇ f f ˇ " and ˇ f fˇ"
ˇ a a
ˇ ˇ a a ˇ
and finally
Z b Z x Z b
f D lim f D f: t
u
x!b a a
a
1.3 Theorem. Let aij .x/; bi .x/ be continuous on an interval J , let x0 2 L and let
j , j D 1; : : : ; n, be arbitrary real numbers. Then the LDE system
X
n
yi0 .x/ D aij .x/yj .x/; i D 1; : : : ; n
j D1
Proof. Uniqueness follows from the general Theorem 4.1 of Chapter 6, from which
we also know that there exists a solution defined on a neighborhood of the point
x0 . We will prove that this solution can be extended on the whole of J . We will
construct the extension on the part of the interval to the right of x0 , the extension to
the left is analogous.
Recall 2.1 of Chapter 6 and denote by M the set of all z 2 J , z x0 such that
there is a solution of the equations
Z z X
n
yi .x/ D . aij .t/yj .t/ C bi .t//dt C i
x0 j D1
on hx0 ; zi. Set s D sup M . If the set M is not all of J \ hx0 ; C1i, we have
(1) s finite, and
(2) s 2 J X M .
((1) is obvious; regarding (2), either s < sup J and there is a solution in one of
its neighborhoods, or s … M while s 2 J , since it is the only point at which
J \ hx0 ; C1i can differ from M ).
Since aij and bi are continuous functions defined on hx0 ; si, they are bounded on
this interval, say
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178 7 Systems of Linear Differential Equations
nAC ˛
e C B. xO / C ji j < C e˛ ;
˛
so is not only in M , but in fact in MQ .
Therefore, MQ D M . Now we will take advantage of the fact that in our
procedure, we did not assume to be in M : the supremum point s can be written as
a limit of an increasing sequence of elements from M (equal to MQ ) in contradiction
with s 2 J X M which followed from the assumption that M ¤ J \ hx0 ; C1i. u t
X
n1
y .n/ C aj .x/y .j / .x/ D b.x/
j D1
has precisely one solution on the interval J satisfying the conditions y .j / .x0 / D j
for all j D 1; : : : ; n 1.
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2 Spaces of solutions 179
2 Spaces of solutions
2.1
aij ; bi
are defined on an open interval J . We denote by C.J / the R-vector space of all
continuous functions on J . Further, we denote the vector space
C.J /
C.J / n times
C n .J /:
2.2 Theorem. The system of all solutions of the LDE system (L) constitutes an
affine subset y0 C W of C n .J /, and the system of all solutions of the n-th order
equation (L̃) constitutes an affine subset y0 C W , where the vector subspaces W are
the sets of all solutions of the associated homogeneous equations.
Theorem. The dimensions of (both of) the affine sets from the previous theorem
are n.
Proof. Again, we will prove the statement for the system (L). Let y1 ; : : : ; yp be
solutions of (L-hom) and let p > n. Take an x0 2 J . Then the system of algebraic
linear equations
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180 7 Systems of Linear Differential Equations
has a non-trivial solution ˛1 ; : : : ; ˛n (in fact, the vector space of such solutions has
dimension p n). Set
X
p
yD ˛i yi :
i D1
In particular we have y.x0 / D .0; : : : ; 0/. But we already know such a solution,
namely zero: o D .const0 ; : : : ; const0 /. From uniqueness, it now follows that
Xp
˛i yi D o, i.e. that the system y1 ; : : : ; yn is linearly dependent; hence, the
i D1
dimension of W is at most n. On the other hand consider the solutions yi of
(L-hom) such that yij .x0P / is 1 for i D 1 and 0 otherwise. P
Then we obtain a linearly
independent
P system: if ˛ i y i D o then in particular ˛i yi .x0 / D 0, that is,
˛i ıij D 0 and all the ˛i are zero. t
u
The functions W .y1 ; : : : ; yn /.x/ resp. W .y1 ; : : : ; yn /.x/ are called the Wronski
determinants of the equations in question.
Remark. Note that the latter is in fact a special case of the former obtained from
the standard translation as in 1.2 of Chapter 6.
2.4 Theorem. The following statements are equivalent for a system of solutions
y1 ; : : : ; yn of the system (L) (the interval J is as before):
(1) the solutions y1 ; : : : ; yn are linearly independent,
(2) W .y1 ; : : : ; yn /.x/ ¤ 0 at all x 2 J ,
(3) there exists an x0 2 J such that W .y1 ; : : : ; yn /.x0 / ¤ 0.
Similarly for the system (L̃). If the conditions hold, the system y1 ; : : : ; yn is called a
fundamental system of solutions.
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3 Variation of constants 181
Proof. We will prove the statement for the case (L̃), just for a change.
(1))(2): Suppose (2) does not hold and we have an x0 2 J such that
ˇ ˇ
ˇ y1 .x0 /; : : : ; yn .x0 / ˇ
ˇ 0 ˇ
ˇ y .x0 /; : : : ; yn0 .x0 / ˇ
W .y1 ; : : : ; yn /.x0 / D ˇˇ 1 ˇ D 0:
ˇ
ˇ ::: ˇ
ˇy .n1/ .x /; : : : ; y .n1/ .x ˇ
1 0 n 0/
3 Variation of constants
This is a method which allows us to find the system of solutions of the system
(L) (resp. (L̃)), provided we know a fundamental system of solutions of the system
(L-hom) (resp. (L̃-hom)). Again, the latter is a special case of the former, but in this
case we will present both cases explicitly.
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182 7 Systems of Linear Differential Equations
and hence
X X X X
yij0 D ci0 yij C ci yij0 D ci0 yij C ci ajk yik
i ik
X X X X X
D ci0 yij C ajk ci yik D ci0 yij C ajk y0k
i k i i k
This is easily done using the Cramer rule (Appendix B, 4.2). If we denote by Wi .x/
the Wronskian in which we replace the i -th column by the
0 1
b1 .x/
@ ::: A
bn .x/
we obtain
Wi .x/
ci0 .x/ D
W .y1 ; : : : ; yn /
Consider a basis y1 .x/; : : : ; yn .x/. Let us look for a solution in the form
X
y.x/ D ci .x/yi .x/:
.n/
X
n1
.j /
We have yi .x/ D aj yi D 0. Thus, if we require
j D0
X
ci0 .x/yi .x/ D 0
.k/
(*)
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4 A Linear differential equation of nth order with constant coefficients 183
:::
X .n1/
y .n1/ .x/ D ci .x/yi .x/:
Then we have
X .n/
y .n/ .x/ D ci .x/yi .x/ C b.x/
The requirements (*) and (**) constitute, again, a system of algebraic linear
equations solvable using the Cramer rule (again with the non-zero Wronskian in the
denominator) to obtain ci0 .x/. Finally, take the primitive functions to obtain ci.x/ .
In this and the following section we will consider linear differential equations with
constant coefficients ai , resp. aij . In view of the previous section, it suffices to solve
the corresponding homogeneous equations. If these are solved, the general case can
be computed by variation of constants; note that the right-hand sides b resp bi do
not have to be constant.
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184 7 Systems of Linear Differential Equations
y.x/ D ex :
We have
and hence the equation (*) will be satisfied if (and only if)
4.2
X
k
pj .x/ej x
j D1
Proof. Suppose not. Then among the counterexamples, choose one such that
(a) the maximum of the degrees of the polynomials pj is the least possible, and
(b) the number of the polynomials pj with this maximum degree is the least
possible.
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4 A Linear differential equation of nth order with constant coefficients 185
Here the degree of a constant non-zero polynomial is defined to be 0, and the degree
of the constant zero is defined to be 1. Thus, taking derivative of a non-zero
polynomial decreases the degree by one. We have identically
X
k
pj .x/ej x D 0: (4.2.1)
j D1
X
k X
k
pj .x/ej x C pj .x/j ej x D 0: (4.2.2)
j D1 j D1
Let, say, p1 have the maximum degree. Subtracting (4.2.1) multiplied by 1 from
(4.2.2), we obtain
X
k
p10 .x/e1 x C ..j 1 /pj .x/ C pj0 .x//ej x D 0: (4.2.3)
j D2
Now the degree of the polynomial at e1 x has decreased and none of the other
degrees has increased. Thus, the formula (4.2.3) cannot be a counterexample to the
statement and hence we have to have
From the second equation we immediately see that all the pj with j > 1 are
identically zero (since 1 ¤ j ). The first one immediately yields only that p1
has to be a constant, but C e1 x is zero only if C D 0. t
u
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186 7 Systems of Linear Differential Equations
e1 x ; : : : ; en x :
The problem is, hence, what to do with the complex roots, and how to deal with a
possible multiplicity of some of the roots.
We are dealing with an LDE in real variables. Thus the characteristic polynomial has
real coefficients and consequently each of the roots which is not real is accompanied
with its complex conjugate as another root. That is, if ˇj ¤ 0 in a root
j D ˛j C iˇj
k D ˛j iˇj :
The two complex functions ej x ; ek x are then in our basis replaced by
Replacing eix and eix by linear combinations of cos x and sin x, and vice versa, in
the present context, is justified by Exercise (12) of Chapter 1. We will gain a much
better understanding of this in Chapter 10 below.
Define an operator
X
n1
L.y/ D y .n/ C aj y .j /
j D0
@n y X @j y
n1
L.y/ D n C aj j :
@x j D0
@x
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5 Systems of LDE with constant coefficients. An application of Jordan’s Theorem 187
@ @ @n y @n @y @y
L.y/ D n
C D n
C D L
@ @ @x @x @ @
In particular for y.x; / D ex we have L.y/ D ex p./ and hence
@k y @k x
L.x e / D L
k x
D .e p.//:
@k @k
Thus we obtain k solutions, and if we apply this to all the roots we obtain n
solutions, independent by 4.3, and hence the fundamental system of solutions we
needed.
For a conjugate pair of complex roots ˛ C iˇ, ˛ iˇ we take, of course,
e˛x cos ˇx; xe˛x cos ˇx; : : : ; x k1 e˛x cos ˇx;
e˛x sin ˇx; xe˛x sin ˇx; : : : ; x k1 e˛x sin ˇx:
y0 D Ay: (5.1.1)
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188 7 Systems of Linear Differential Equations
In fact, let us carefully consider two contexts in which (5.1.1) makes sense. The first
context is, as above, when A is a constant n
n matrix over R, and y W R ! Rn is an
unknown vector-valued function. However, it also makes sense to consider the case
when A is an n
n matrix over C, and the unknown function is y W R ! Cn . This
case makes sense since we may identify C Š R2 , and such system of n complex-
valued first order differential equations can therefore be interpreted as a system of
2n real-valued first-order linear differential equations. Let us emphasize, however,
that in this discussion, the independent variable remains real.
The advantage of considering (5.1.1) over C is that over C, every matrix is similar
to a matrix in Jordan canonical form. Changing basis to the basis in which the
matrix is in Jordan form gives a substitution which allows us to solve the system
of equations. Even more explicitly, this can be said as follows: consider a k
k
Jordan block of the matrix A with respect to an eigenvalue . This corresponds to k
vectors u1 ; : : : ; uk 2 Cn such that
Au1 D u1 ;
(5.1.2)
Auj D uj C uj 1 ; j D 2; : : : ; k:
Then this data give the following solutions of the system (5.1.1):
u1 e x ;
u2 e x C u1 xe x ;
(5.1.3)
:::
x k1 x
uk e x C uk1 xe x C C u1 .k1/Š e :
Taking the solutions (5.1.3) for all Jordan blocks gives a fundamental system of
solutions, which we can see by taking the determinant of their values at 0 (where
we get the base change matrix from the Jordan basis to the standard basis); recall
from Theorem 2.4 that a system of n solutions whose values are independent at one
point is a fundamental system of solutions.
5.2
Let us now consider the case when the system (5.1.1) is over R. Then, the matrix A
is a real matrix. This means that for every solution y over C,
are real solutions of (5.1.1). Taking all such solutions for all Jordan blocks gives a
system of real solutions which, when considered over C, generate the vector space
of all the complex solutions and hence must contain a basis of the space of real
solutions (which can be found explicitly by finding a set of columns which form a
basis of the matrix of values at 0).
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5 Systems of LDE with constant coefficients. An application of Jordan’s Theorem 189
5.2.1 Example
Consider the system (5.1.1) with
0 1
0 1 0 0
B1 0 0 0C
ADB
@0
C:
0 0 1 A
1 0 1 0
A .x/ D .x 2 C 1/2 ;
u1 D .0; 0; 1; i /T ;
u2 D .2i; 2; 0; 1/T :
Note that we could equivalently take a scalar multiple of both vectors by the same
non-zero complex number. Thus, (5.1.3) produces solutions
.0; 0; e ix ; i e ix /T ;
.2i; 2; 0; 1/T e ix C .0; 0; 1; i /T xe ix :
Since the data obtained from the other Jordan block can be taken complex conjugate,
we know that these solutions span the space of all complex solutions, and hence
form a fundamental system of real solutions.
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190 7 Systems of Linear Differential Equations
5.3 Remark
y .n/ C a1 y .n1/ C C an y D 0
We clearly have
6 Exercises
(1) Prove that the Wronskian W .x/ of any n solutions of the system
y0 D A.x/y
(Here for a square matrix A, tr.A/ is the sum of its diagonal terms.)
(2) The differential equation
y0 y
y 00 C D0
x x
has solutions
1
y D x; y D :
x
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6 Exercises 191
y0 y
y 00 C D ex :
x x
(3) Find a fundamental system of solutions of the equation
y10 D y1 y2 C xe x ;
y20 D y1 C 3y2 C x 2 :
(5) Find a fundamental system of real solutions of the system of LDE’s (5.1.1)
with
0 1
1 1 0 1
B 1 1 0 1C
ADB
@ 0
C:
0 1 1A
1 0 1 1
y0 D Ay:
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In this chapter, we introduce the line integral and prove Green’s Theorem which
relates a line integral over a closed curve (or curves) in R2 to the ordinary integral of
a certain quantity over the region enclosed by the curve(s). Making rigorous sense
of what this last concept means is a big part of the work. Much of the material
of this section is subsumed by the more general treatment of Stokes’ Theorem
in manifolds of arbitrary dimension in Chapter 12 below. However, there are two
important reasons to present Green’s Theorem first. The first reason is that Green’s
Theorem is much more elementary, and does not require the added abstraction, and
algebra and topology material needed for Stokes’ Theorem. The other important
reason is that Green’s Theorem can be, in fact, used directly to set up the foundations
of basic complex analysis, which we do in the next chapter, and which is rather nice
to do without having to go into Stokes’ Theorem in a general dimension.
1.1
D .1 ; : : : ; n /T W ha; bi ! Rn
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194 8 Line Integrals and Green’s Theorem
1.2
W ha; bi ! Rn ;
W ha; bi ! Rn
;
ı ˛ D :
Note that the relation
really is an equivalence relation, i.e. that it is reflexive,
symmetrical and transitive (see Exercise (2)). Equivalence classes with respect to
will be called piecewise continuously differentiable curves.
1.3 Remark
Clearly,
implies Œha; bi D Œhc; d i. On the other hand, if and are
one-to-one and we have Œha; bi D Œhc; d i, then we have
. In effect,
consider the maps
defined by the same formulas as ; . Since the relevant spaces are compact, ,
1
are homeomorphisms (see 6.2.2 of Chapter 2). Put ˛ D :
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1 Curves and line integrals 195
1.5
W ha; bi ! Rn ; W ha; bi ! Rn
.t/ D t C b C a:
The geometric idea of a curve is modelled well by the concepts of 1.2 (see 1.3).
A parametrization can be interpreted as additional information about “time” at
which we are at a particular point when travelling along the curve. In an oriented
curve, we do not care about the precise time at which we are at a particular point,
but we do want to keep track of the direction of travel.
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196 8 Line Integrals and Green’s Theorem
W ha; bi ! Rn :
1.7
L C K:
1.8
L
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2 Line integrals of the first kind (D according to length) 197
2.2
Recall the definition of the Riemann integral and view it informally as a kind of
summation of the function f over the length of an interval. Since an interval is
a very special case of a piecewise continuously differentiable curve (where the
parametrization is the identity), we may wonder if the Riemann integral could
be generalized to a situation where the domain is (the image of) a piecewise
continuously differentiable curve. This intuition indeed works. By a partition of
a parametrized piecewise continuously differentiable curve W ha; bi ! Rn we
will mean a sequence of points
where t0 < t1 < < tk is a partition of the interval ha; bi. The mesh of a partition
is the maximum of the numbers jj.ti 1 / .ti /jj.
Note that since ha; bi is a compact space, is a uniformly continuous map and
hence if the mesh of a sequence of partitions goes to 0, so does the mesh of their
-images.
Now consider a continuous real function f defined (at least) on Œha; bi. In
analogy with the Riemann integral, (recall, in particular, Theorem 8.3 of Chapter 1),
let us investigate sums of the form
X
k
f ..ti //jj.ti / .ti 1 /jj
i D1
and let us see if they converge to a particular value when the mesh goes to 0. By the
Mean Value Theorem
v
X uX
u n
f ..ti //t .j .ti / j .ti 1 //2
i j D1
X sX
D f ..ti // j0 .ij /2 .ti ti 1 /2
i j
X sX
D f ..ti // j0 .ij /2 .ti ti 1 /
i j
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198 8 Line Integrals and Green’s Theorem
the line intergral of the first kind (or integral according to length) of the function f
over the curve L, and denote it by
Z Z
f or f .x/jjdxjj:
L L
2.2.1 Comment
The formula (**) makes sense, of course, for any integrable function f , in which
case the integral (**) exists as a Lebesgue integral. A similar comment will apply
to all the types of curve integrals we shall introduce. It is useful to note, however,
that in the context of the present chapter, we are not interested in such level of
generality, and are happy to assume that the function f is continuous in which case
the Lebesgue integral is the same as the Riemann integral. Nevertheless, even with
that in mind, the Lebesgue integral techniques we developed in Chapter 5 are still
needed for example in arguments such as differentiating behind the integral sign in
Proposition 3.7 or the use of multivariable substitution in Section 5 below.
2.3 Proposition. The expression in the definition of the line integral of the first kind
is independent of parametrization.
qP
0 2 0 0
D j .˛.t// j˛ .t/j D jj .˛.t//jj j˛ 0 .t/j
and hence by the Substitution Theorem (for the Riemann integral in one variable),
we have
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3 Line integrals of the second kind 199
Z b Z b
f ..t//jj0 .t/jjdt D f . .˛.t//jj 0
.˛.t//jj j˛ 0 .t/jdt
a a
Z d
0
D f . .//jj ./jjd: t
u
c
(The attentive reader will recall from the theory of the single variable Riemann
integral substitution that if ˛ is decreasing, the absolute value in j˛ 0 .t/j is
nevertheless correct because of an interchange of bounds.)
2.4 Remark
The length of a curve L is defined as the integral of the first kind of the function 1
over L, i.e.
Z Z b
1D jj0 jj:
L a
3.1
(note, in the middle expression, the dot product of vectors). When there is a danger
of confusion, we will denote line integrals of the first and second kind explicitly by
Z Z
.I / ; .II/ :
L L
In the literature, the line integral of the second kind is also often denoted by
Z
.f1 dx1 C C fn dxn /:
L
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200 8 Line Integrals and Green’s Theorem
This notation, in fact, conforms to the notation of differential forms, which we will
see later in Chapter 12. When x D .x1 ; : : : ; xn /T , we will also use the notation
Z
f.x/ dx:
L
3.2
3.3 Proposition. The expression in the definition of the line integral of the second
kind does not depend on the choice of parametrization of an oriented piecewise
continuously differentiable curve.
Proof. Let D ı ˛. Now, of course, ˛ 0 .t/ > 0 (with the possible exception of
finitely many points, where ˛ 0 has, at most, discontinuities of the first kind). We have
n Z
X b n Z
X b
fj ..t//j0 .t/dt D fj . .˛.t// 0 0
j .˛.t//˛ .t/dt
j D1 a j D1 a
n Z
X d
0
D fj . .// j ./d: t
u
j D1 c
Z Z
Observation. f D f.
L L
3.4
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3 Line integrals of the second kind 201
3.5
0 .t/
f..t// D f ..t// :
jj0 .t/jj
Thus, the line integral of the first kind can be reduced to the line integral of the
second kind.
3.6 Remarks
1. The traditional terms “of the first kind” and “of the second kind” therefore should
not be interpreted as expressing the order of importance. The line integral of the
second kind is in fact more fundamental, and the integral of the first kind can be
reduced to it. Perhaps the reason for the terminology is that the line integral of
the first kind is the more naive notion.
2. The function f or f often is defined on an open set containing Œha; bi. This
will play a crucial role in the proof of Green’s Theorem.
3.7
3.8 Proposition. Let f.˛; x/ be a continuous vector function defined in an open set
@fj .˛; x/
U of Rn such that is continuous on U for each j . Then the line integral
@˛
of the second kind satisfies
Z Z
d @f.˛; x/
f.˛; x/ dx D dx:
d˛ L L @˛
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202 8 Line Integrals and Green’s Theorem
4.1
For a complex function of one real variable, f .t/ D f1 .t/ C if2 .t/ where f1 , f2 are
real functions, one introduces the Riemann integral by the formula
Z b Z b Z b
f .t/dt D f1 .t/dt C i f2 .t/dt:
a a a
4.2
Recall that on the field of complex numbers C, we use the distance function
d.x; y/ D jx yj, which is the same as the Euclidean distance when we identify
C with R2 by x C iy 7! .x; y/. We will use this identification freely to define
piecewise continuously differentiable functions in C, etc., but now note that .t/
are the elements of the field C and hence can be subjected to the multiplication in C
which is different from the dot multiplication in R2 (for example in that the result is
again an element of C rather than R). This distinction, in fact, is the main point of
the present section. Because of this, when working with complex-valued functions,
we will not use bold-faced letters as we did in the case of vector functions.
4.3
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4 The complex line integral 203
4.4
It is, however, again possible to express the complex line integral in terms of line
integrals of the second kind.
where f1 , f2 are real functions of one complex variable. Then the complex line
integral satisfies
Z Z Z
f .z/dz D .II/ .f1 ; f2 /T C i .II/ .f2 ; f1 /T :
L L L
Proof. We have
Z b Z b
f ..t// 0 .t/dt D .f1 ..t// C if2 ..t///.10 .t/ C i 20 .t//dt
a a
Z b
D .f1 ..t//10 .t/ C .f2 ..t///20 .t//dt
a
Z !
b
Ci .f2 ..t//10 .t/ C f1 ..t//20 .t/dt
a
Z Z
D .f1 ; f2 /T C i .f2 ; f1 /T : t
u
L L
Remark: This theorem also implies that the complex line integral does not
depend on the parametrization of an oriented piecewise continuously differentiable
curve. (Of course, reversal of orientation results in a reversal of sign.)
4.5
The estimate in the following statement is not particularly tight. However, it will
prove useful in Chapter 10 below.
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204 8 Line Integrals and Green’s Theorem
Proof. We have
ˇZ ˇ ˇZ Z b Z b Z b ˇ
ˇ b ˇ ˇ b ˇ
ˇ 0 ˇ ˇ 0 0 0 0ˇ
ˇ f ..t// .t/dt ˇ D ˇ f1 1 f2 2 C i f2 1 C i f1 1 ˇ
ˇ a ˇ ˇ a a a a ˇ
ˇZ ˇ ˇZ ˇ ˇZ ˇ ˇZ ˇ
ˇ b ˇ ˇ b ˇ ˇ b ˇ ˇ b ˇ
ˇ 0ˇ ˇ 0ˇ ˇ 0ˇ ˇ 0ˇ
ˇ f1 1 ˇ C ˇ f2 2 ˇ C ˇ f2 1 ˇ C ˇ f1 2 ˇ
ˇ a ˇ ˇ a ˇ ˇ a ˇ ˇ a ˇ
Z b Z b Z b Z b
jf1 j j10 j C jf2 j j20 j C jf2 j j10 j C jf1 j j20 j
a a a a
Z b Z b
4 Aj 0 j D 4A j 0 j D 4Ad: t
u
a a
5 Green’s Theorem
Let Z Rn be a compact set, and let S be a set of open subsets of Rn whose union
contains Z. A smooth partition of unity subordinate to S is a set of finitely many
smooth functions i W Rn ! R, i D 1; : : : ; k such that the image of each i is
contained in h0; 1i, the support of each i is compact and contained in one of the
Xk
sets from S , and D i has the property that .x/ D 1 for every x 2 Z.
i D1
Lemma. Let Z Rn be a compact set. For every set S whose union contains Z,
there exists a smooth partition of unity.
Consider further the functions i;A .x/ D .xi Bi /, i;B .x/ D .Ai xi /, and let
be the sum of all these functions. Then J D J =, J 2 F form a smooth partition
of unity. t
u
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5 Green’s Theorem 205
5.2
x W Vx !
.o; 1/ D fx 2 R2 j jjxjj < 1g
with
det.Dx / > 0;
[
k
. Im.cj // \ Vx D ci Œ.b a; b C a/;
j D1
(4) For s 2 h1; 0i, we have x ci .as C b/ D .s cos.˛/; s sin.˛// and for s 2
h0; 1i, we have x ci .as C b/ D .s; 0/:
5.3 Comment
Informally, the above definition says simply that the boundary of U is a union of the
images of the Li ’s and that at a neighborhood of every point of the boundary, locally
U looks like a wedge of an open disk (the wedge may also be a half-disk) whose
boundary is parametrized linearly by one of the curves ci in the same direction as
the increasing parametrization of .1; 1/ is with respect to the upper half-disk
f.x; y/ 2
.o; 1/jy 0g:
Note, however, the great generality this allows, for example a disk D with several
open disks removed whose disjoint closures are in the interior of D, or similarly
with polygons, etc. The beauty of the upcoming proof is that it uses no intuitive
properties of such situations except the formal properties given in the definition;
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206 8 Line Integrals and Green’s Theorem
for example, we do not use any intuitive notion of “interior” or “exterior” of the
curves Li , and although the expression “counter-clockwise” matches the intuition,
Definition 5.2 is not based on intuition. Another way of putting this is to note that
our definition of boundary is purely local in the sense that it is completely described
by requirements on neighborhoods of individual points of C.
5.4
Let
Z Z Z
fD f CC f:
L1 qqLk L1 Lk
c1 W h0; 4i ! R2
defined by
In this case, applying Fubini’s Theorem and the Fundamental Theorem of Calculus
in one variable, we get
Z Z K Z K
@f2 @f2 .x1 ; x2 /
D dx1 dx2
M @x1 0 0 @x1
Z K Z Z
D .f2 .K; x2 / f2 .0; x2 // dx2 D fC f:
0 L1 L3
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5 Green’s Theorem 207
Similarly, we have
Z Z K Z K
@f1 @f1 .x1 ; x2 /
D dx2 dx1
M @x2 0 0 @x2
Z K Z Z
D .f1 .x1 ; 0/ f1 .x1 ; K// dx1 D fC f:
0 L4 L2
Adding these two formulas gives the statement in the present case. Amazingly, this
is the only concrete case of the theorem we need to prove by direct calculation.
Now consider the general case. First we need to observe that the statement (5.4.1)
doesn’t change if we perform a (2-variable) substitution by a diffeomorphism W
V ! V 0 (see Theorem 7.9 of Chapter 5). This is easy to accept, but somewhat
harder to do in detail. The reason is that even in two variables, the concepts we set
up so far do not transform in the simplest possible way under coordinate change.
We will understand this better in Chapter 12 below.
To do the calculation we need, let us write
.x1 ; x2 /T D F..r1 ; r2 /T /;
so identifying, at a point, the linear map D with its associated matrix, we have
0 @x @x 1
1 1
B @r1 @r2 C
B C
DF D B C:
@ @x @x A
2 2
@r1 @r2
(Note that if we wrote the integrand of a line integral of the second kind as a row
instead of column vector, the transposition on the right hand side of (5.4.2) would
be unnecessary - again, we will understand this better in Chapter 12 below.)
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208 8 Line Integrals and Green’s Theorem
Now compute:
@g2 @g1
@r1 @g2
We see that the second order terms cancel out, and after applying the chain rule
the right hand side of (5.4.3) becomes a sum of eight terms, four of which cancel
out, leaving
@f2 @f1 @x1 @x2 @x1 @x2 @f2 @f1
D det.DF/;
@x1 @x2 @r1 @r2 @r2 @r1 @x1 @x2
V1 [ [ Vm U
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6 Exercises 209
Z D .h1; 1i
h1; 1i/ X ..0; 1i
h1; 0//:
Express
Z D Z1 [ Z2
where
Z1 D h1; 1i
h0; 1i;
Z2 D h1; 0i
h1; 0i:
The sets are not disjoint, but the intersection has measure 0. For the sets Z1 , Z2
and restrictions of the function ui f ı .x /1 , the statement follows from Cases 3
and 2, respectively. When adding the left hand sides of formula (5.4.1) for these
functions, the contributions from the line segment h1; 0i
f0g cancel out. u t
6 Exercises
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210 8 Line Integrals and Green’s Theorem
0.t /
(6) Prove directly that the factor of 3.4 at each point Œha; bi (where
jj0.t / jj
defined) does not depend on the parametrization of the piecewise continuously
differentiable oriented curve. Prove also that reversal of orientation of the
curve results in multiplication of this factor by 1.
(7) Compute the complex line integral
Z
e z dz
L
oriented counterclockwise.
(10) Prove that if L1 q q Lk is the boundary of a domain U oriented
counter-clockwise, then the area of U is equal to
Z
1
xdy ydx:
2 L1 qqLk
oriented counter-clockwise.
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Part II
Analysis and Geometry
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For the remaining chapters of this text, we must revisit our foundations. Specifically,
it is time to upgrade our knowledge of both metric and topological spaces. For
example, in the upcoming discussion of manifolds in Chapter 12, we will need
separability. We will need a characterization of compactness by properties of open
covers. Also, it is natural to define manifolds as topological and not metric spaces
which prompts the development of separation axioms, with a focus on normality. On
the other hand, when discussing Hilbert spaces in Chapters 16 and 17, we will need
completion, extension of uniformly continuous maps, and the Stone-Weierstrass
Theorem. These are the topics we will discuss in the present chapter.
1.2 Theorem. The following statements about a metric space X D .X; d / are
equivalent.
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214 9 Metric and Topological Spaces II
(1) X is separable.
(2) The topology of X has a countable basis.
(3) X is Lindelöf.
B D f
.m; r/ j m 2 M; r rationalg:
We will prove that B is a basis. Take an open U , an x 2 U , and an " > 0 such that
.x; "/ U . Now choose an m 2 M such that d.x; m/ < 13 " and a rational r such
that 13 " < r < 23 ". Then x 2
.m; r/
.x; "/ U : in effect, if d.m; y/ < r we
have d.x; y/ d.x; m/ C d.m; y/ < . 13 C 23 /" D ".
(2))(3): Let B be a countable basis and let U be an arbitrary open cover. Put
B 0 D fB 2 B j 9U 2 U; B U g. Then B 0 is a countable cover, and if we choose
for each B 2 B 0 a UB 2 U with B UB then also fUB j B 2 B 0 g is a countable
cover.
(3))(1): For every positive natural number n, choose a countable subcover of
the cover f
.x; n1 / j x 2 X g, say
.xn1 ; n1 /; : : : ;
.xnk ; n1 /; : : : :
1.2.1 Remarks
1. This is a very specific fact concerning metric spaces. In a general topological
space one has only the (very easy) implications (2))(3) a (2))(1) and nothing
more.
2. In the literature, the existence of a countable basis is often called the second
axiom of countability.
1.2.2
Obviously, if X has a countable basis B then each subspace Y X has one, namely
BjY D fU \ Y j U 2 Bg. Hence we have
The first of these statements hardly comes as a surprise (it is easy to prove it
directly, too). But the second one should sound somewhat strange. We will see
shortly (in 2.3 below) that Lindelöf property is very close to compactness, and
compactness is (very obviously) not preserved on subspaces. Again, this corollary is
characteristic for metric spaces. In a general topological context neither of the two
statements holds.
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1 Separable and totally bounded metric spaces 215
1.3
A metric space X is said to be totally bounded if for each " > 0 there exists a finite
subset M."/ of X such that
1.3.1
A totally bounded space is always bounded but a bounded space is not necessarily
totally bounded: take any infinite set and define d.x; y/ D 1 for x ¤ y. But we
have
X hN; N i
hN; N i
N "
for a suficiently large natural N . Choose a natural number k such that k
< 2
and put
For every x 2 X , there exists an s 2 M such that d.x; s/ < 2" . For an s 2 M ,
choose an x.s/ 2 X such that d.x.s/; s/ < 2" , if such x.s/ exists, and put
Then, by the triangle inequality, we have, for every x 2 X , d.x; MX / < "
2 C 2" D ".
t
u
1.4 Proposition. A metric space X is totally bounded if and only if every sequence
in X contains a Cauchy subsequence.
Proof. I. Let X be totally bounded. Consider the sets M. n1 / from the defini-
tion 1.3. Now consider a sequence .xi /i D1;2::: in X . If the set P D fxi j i D
1; 2; : : : g is finite, then our sequence contains a constant subsequence, which is,
of course, Cauchy. Otherwise choose first m1 2 M.1/ so that P1 D P \
.m1 ; 1/ is infinite, and then k1 with xk1 2 P1 . Now assuming we have
mj 2 M. j1 /; j D 1; : : : ; n 1; such that Pj D Pj 1 \
.mj ; j1 /
are infinite, and k1 < k2 < < kn1 such that xkj 2 Pj ;
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216 9 Metric and Topological Spaces II
2.1
2.1.1 Proposition. A metric space X is compact if and only if every infinite set
M X has an accumulation point.
2.2 Theorem. A metric space X is compact if and only if it is complete and totally
bounded.
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2 More on compact spaces 217
2.2.1 Remark
This fact is a generalization of Theorem 6.5 of Chapter 2 stating that a subset X
Rn is compact if and only if it is closed and bounded. We know that Rn is complete
(7.5 of Chapter 2), and hence, by 7.5 of Chapter 2 again, X is complete if and
only if it is closed; by 1.3.1, for X Rm , boundedness and total boundedness are
equivalent.
2.3
Theorem. A metric space is compact if and only if each of its (open) covers
contains a finite subcover.
Proof. Let X be compact and let U 0 be a cover of X which has no finite subcover.
By 2.2 and 1.5, X is separable, hence by 1.2 it is Lindelöf, and hence U 0 has a
countable subcover
U D fU1 ; U2 ; : : : ; Un ; : : : g:
[
n1
let Vn D Uj for the lowest j such that Uj X Vi ¤ ;
i D1
(by assumption, the finite system fV1 ; : : : ; Vn1 g cannot be a cover). Choose
[
n1
xn 2 Vn X Vi and put M D fxn j n D 1; 2; : : : g:
i D1
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218 9 Metric and Topological Spaces II
Now
[
n1
• xn … fx1 ; : : : ; xn1 g . Vi / and hence M is infinite,
i D1
• V1 ; : : : ; Vn ; : : : is a cover since each discarded Uj is contained in the union of
the Vi ’s, and
• Vn \ M fx1 ; : : : ; xn g and hence is finite.
This is a contradiction: The set M must have an accumulation point x, this x is an
element of some Vn , but this neighborhood of x meets M in finitely many points
only.
II. Assume each cover of X has a finite subcover and assume M X has no
accumulation point. Then for every x 2 X there exists an open neighborhood Un
such that Un \ M is finite. Choose a finite subcover Uk1 ; : : : ; Ukn . Then
[
n [
n
M D. U xi / \ M D .Uxi \ M /
i D1 i D1
2.4
Theorem 2.3 suggests the following definition of compactness for general topologi-
cal spaces, which we will adopt from now on:
A topological space is said to be compact if each of its (open) covers has a finite
subcover.
Similarly as in the special case of metric spaces (recall 6.2 of Chapter 2) we have
[
n [
n
X f 1 ŒUi D f 1 Œ Uij :
j D1 j D1
[
n
This is equivalent to f ŒX U ij . t
u
j D1
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3 Baire’s Category Theorem 219
2.4.4 Remark
Unlike the case of metric spaces, a compact subspace of a topological space is
not necessarily closed: for example, any subspace of a finite topological space is
compact, but not every subset may be closed. This, in fact, is one of the motivations
of separation axioms, which can be used to remedy this situation, and which will be
discussed in Section 5 below.
3.3
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220 9 Metric and Topological Spaces II
S
1
A subset A X is of the first category in X if it is a union An of an increasing
nD1
sequence A1 A2 of nowhere dense subsets.
Proof. Let
A1 A2 An
.x2 ; 2"2 / \ A2 D ; and
.x2 ; 2"2 /
.x1 ; "1 /:
Now assume we already have x1 ; : : : ; xn and "1 ; : : : ; "n , 0 < "k < k1 , such that
.xk ; 2"k / \ Ak D ; for k n; and
.xk ; 2"k /
.xk1 ; "k1 / for 1 < k n:
Since
.xn ; "n / is a non-empty open set, we have a non-empty open
.xn ; "n / \
.X X AnC1 / and hence there is an xnC1 and an "nC1 with 0 < "nC1 < nC1
1
such that
.xnC1 ; 2"nC1 / \ AnC1 D ; and
.xnC1 ; 2"nC1 /
.xn ; "n /:
Since
.x; "/
.x; 2"/ (if d.y;
.x; "// D 0 we can find a z 2
.x; "/ such that
d.y; z/ < "), setting Bn D
.xn ; "n / we obtain a sequence
.x1 ; 2"1 / B1
.x2 ; 2"2 / B2
.x3 ; 2"3 / B3
such that
.xk ; 2"k / \ Ak D ; (and hence Bk \ Ak D ;).
For k n we have xk 2
.xn ; 2"n / and since "n < n1 the sequence .xn /n is
Cauchy, and by completeness it has a limit x 2 X . Furthermore,
T for k n we have
xk 2 TBn , and since Bn is closed, xT2 Bn . Thus,
S x 2 Bn . Since Bn \SAn D ; we
have Bk \ An D ; and finally Bk \ An D ;. Therefore, x … An . t
u
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4 Completion 221
4 Completion
4.1
4.2 Lemma. 1. If .xn /n and .yn /n are Cauchy sequences in X then .d.xn ; yn //n
is a Cauchy, and hence convergent, sequence in R.
2. If .xn /n
.xn0 /n and .yn /n
.yn0 /n then limn d.xn ; yn / D limn d.xn0 ; yn /.
Thus, if d.xm ; xn /; d.ym ; yn / < 2" , then jd.xm ; ym / d.xn ; yn /j < ".
2. d.xn ; yn / d.xn ; xn0 / C d.xn0 ; yn0 / C d.yn0 ; yn / and hence lim d.xn ; yn /
lim d.xn0 ; yn0 /, and by symmetry also lim d.xn0 ; yn0 / lim d.xn ; yn /. t
u
4.3
Denote by XQ the set of all the
-equivalence classes of Cauchy sequences in .X; d /.
For ; 2 XQ , define
Q dQ / is a metric space.
Observation. XQ D .X;
4.4
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222 9 Metric and Topological Spaces II
(Note that (*) implies that f is one-to-one. Thus, to verify that a mapping satisfying
this condition is an isometry it suffices to prove that it is onto.)
If such a mapping exists we say that the spaces .X; d / and .X 0 ; d 0 / are isometric.
A map satisfying the condition (*) without assuming that it is onto will be called
an isometric embedding.
x; x; x; : : : :
Q W X ! X D fxQ j x 2 X g XQ
D .x 7! x/
is an isometry.
I. X is dense in XQ . Consider an arbitrary " > 0. For a 2 XQ , choose a
representative .xn /n and an n0 such that d.xm ; xn / < " for m; n n0 . Then
dQ .n ; xQ n / < n1 :
For an " > 0, choose an n0 such that dQ .m ; n / < " whenever m; n n0 . Then
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4 Completion 223
4.5
(by the isometric embedding requirement, .'.xn //n is Cauchy and hence convergent
in Y ; if .xn /n
.yn /n , then again by the isometric embedding requirement,
and hence lim '.xn / D lim '.yn / so that the definition does not depend on the
n n
choice of a representative).
We have f .x/Q D '.x/ (the limit of a constant sequence), and since a metric is
(obviously) a continuous function, we have
When discussing the Fourier transform in Chapter 17, we will need the following
important result on extension of uniformly continuous maps to the completion.
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224 9 Metric and Topological Spaces II
Then
t
u
Topological spaces are seldom used in the generality of Chapter 2, Section 4. For
various purposes, extra assumptions are usually added. In analysis, we typically
encounter so-called separation axioms, (in fact, typically, the stronger ones), which
we will briefly introduce in this section. It is worth noting that in this context,
separation refers to separation of points or subsets by open sets; it is not related
to separability as defined in Section 1 above.
5.1 T0 and T1
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5 More on topological spaces: Separation 225
It should be noted that while there is not much use for spaces that are not T0 ,
spaces which are not T1 are used a lot (typically, however, in applications outside
analysis).
A space is Hausdorff (or, T2 ) if for any two distinct points x; y 2 X there are
disjoint open sets U; V such that x 2 U and y 2 V .
Hausdorff spaces are already “analysis-friendly”; for instance they admit con-
cepts of convergence in which limits are unique. We will not discuss such topics but
will present the following fact which has been promised before.
5.3.1 Proposition. A topological space X is regular if and only if for every open
U X,
[
U D fV j V open; V U g:
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226 9 Metric and Topological Spaces II
Proof. I. Let X be regular and let x 2 U . Then x … X X U and there are disjoint
open sets V 3 x and W X X U . Now V X X W U and since X X W
is closed, V U .
II. Let the condition hold, let A be closed, and let x … A. Then
[
x2 fV j V open; V X X Ag
5.4 Normality
A space is normal (or T4 ) if for any two disjoint closed subsets A; B X , there
exist disjoint open sets U; V such that A U and B V .
5.4.1 Remarks
1. After 5.3, the reader may expect an axiom T4C 1 requiring a separation of disjoint
2
closed sets by continuous real functions. This, however, already follows from
normality as we will see in 5.4.6 below. On the other hand, complete regularity
does not follow from regularity.
2. Of course we have T2 ) T1 ) T0 while we do not have such implications for
the higher separation axioms (T3 does not imply T2 , T4 does not imply T3 ). The
reason is that the higher separation axioms in fact do not require that points
be closed. In practice, one usually works with T3 &T1 , T3C 1 &T1 and T4 &T1
2
and then the expected implications from “higher” to “lower” separation axiom
naturally hold.
Proof (Recall 8.4 of Chapter 2). For disjoint closed sets A; B X define a maping
' W X ! h0; 1i
by setting
d.x; A/
'.x/ D :
d.x; A/ C d.x; B/
Since the A; B are closed and disjoint we cannot have simultaneously d.x; A/ D 0
and d.x; B/ D 0 and hence d.x; A/ C d.x; B/ > 0 for all x . Thus, ' is continuous
and we can take U D ' 1 Œh0; 12 / and V D ' 1 Œ. 12 ; 1i . t
u
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5 More on topological spaces: Separation 227
Proof. Let X be regular Lindelöf and let A; B be closed and disjoint sets. For a 2 A,
choose open disjoint sets Ua 3 a and Va0 B.
fUa j a 2 Ag [ fX X Ag is a cover of X and therefore we have a subcover
X X A; U1 ; : : : ; Un ; : : : :
U1 U2 Un :
Now set
[
n [
UQ n D Un X V j; U D UQ n ; and
j D1 n
[
n [
VQn D Vn X Uj; V D VQn :
j D1 n
We have A [
U (no point of A appears in any of the subtracted V j ) and B V ,
and U \ V D .UQ m \ VQn / D ;, since in any of the intersections Um \ VQn , we have
m;n
either m n or m n. t
u
Proof. By 5.4.3, it suffices to prove that the space is regular. Let A be closed and
x … A. For a 2 A choose disjoint open sets Ua 3 a and Va 3 x. Then fUa j a 2 Ag
[
n
is a cover of A and hence there is an open subcover Ua1 ; : : : ; Uan . Set U D Uai
i D1
\
n
and V D Vai . Then x 2 V , A U and U \ V D ;. t
u
i D1
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228 9 Metric and Topological Spaces II
5.4.5 Lemma. Let Q h0; 1i be a dense subset. Let us have in a topological space
X open sets Uq , q 2 Q, such that
q<r ) U q Ur :
'.x/ D inffq j x 2 Uq g:
x … Uq ) '.x/ q: (*)
For q < '.x/ take an r with q < r < '.x/; then x … Ur and we see that
Let '.x/ 2 .˛; ˇ/ (the cases '.x/ D 0 or 1 are only simpler and can be left to the
reader). Choose ˛ < q < ' < r < ˇ. Then by the implications above,
Thus, the neighborhood Ur X U q of x is being mapped into .˛; ˇ/ and we see that
' is continuous. t
u
Proof. Let Q be the set of all dyadic rationals between 0 and 1, that is, the
k
; n D 1; 2; : : : I k D 1; 2; : : : ; 2n 1:
2n
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6 The space of continuous functions revisited: The Arzelà-Ascoli Theorem and : : : 229
U. 2kn / U. 2kC1
2nC1
/ and X X U. kC1
2n
/V (and hence U. 2kC1
2nC1
/ U. kC1
2n
//
where for k D 0 we take the set A instead of U.0/ and for k D 2n we take B instead
of X X U.1/.
Thus we obtain inductively a system U.q/, q 2 Q, satisfying the requirements
of Lemma 5.4.5, and the statement follows. t
u
5.4.7 Remarks
1. In particular, every Lindelöf regular space is completely regular. It should be
noted that, with the exception of T3 » T3C 1 , proving that a lower separation
2
axiom does not imply a higher one is easy. This exception, on the contrary,
was a hard nut to crack (and had been an open problem for quite some
time). Proposition 5.4.6 shows why: the counterexample has to use uncountable
reasoning in a substantial way.
2. Lemma 5.4.5 can be used to reformulate complete regularity without referring to
the real numbers. Recall 5.3.1. Denote by the relation
V U df V U:
Certain very strong theorems hold about the space C.K/ of (necessarily bounded)
continuous real functions on a compact metric space K with the supremum metric
considered in 7.7 of Chapter 2. We will prove two such results in the this section,
and use them in Chapters 10 and 17 below.
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230 9 Metric and Topological Spaces II
Thus, this means that the functions fn are all uniformly continuous with the same
bound ı depending on ", independent of n.
Proof. By Theorem 2.2, the space K is totally bounded. Therefore, for each " 2 N,
there is a finite subset S" K such that for every x 2 K, d.x; y/ < " for at least
one y 2 S" .
Now let
[
SD S1=k D fx1 ; x2 ; x3 ; : : : g:
k
6.3 Theorem. Suppose that X is a -compact metric space. Then every sequence
.fn /n in C.X / which is equicontinuous and bounded on every K X compact has
a subsequence which is uniformly convergent on every K X compact.
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6 The space of continuous functions revisited: The Arzelà-Ascoli Theorem and : : : 231
for Kn compact. Then using Theorem 6.2, choose a subsequence .fi1n /n which
converges uniformly on K1 . Within this subsequence, choose another subsequence
.fi2n /n which converges uniformly on K2 . Proceeding in the same way, keep
choosing consecutive subsequences, so that .fij n /n converges uniformly in Kj .
Then the “diagonal” subsequence .finn /n satisfies the requirement. t
u
Notice that the space C.K/ has the structure of a vector space over R, and that the
operations of addition and multiplication by a scalar are continuous. In addition to
this, C.K/ also has an operation of product of function, which is also continuous.
We will consider subsets A C.K/ satisfying the following assumptions:
(1) A is a vector subspace of C.K/, contains the constant function 1 with value 1,
and for f; g 2 A, we have f g 2 A. (We say that A is a unital subalgebra of
C.K/.)
(2) For any two points x; y 2 K, there exists a function f 2 A such that f .x/ ¤
f .y/ (we say that A separates points).
The proof of this theorem will occupy the remainder of this section. However,
let us observe one thing right away: since the operations of addition of functions,
multiplication of functions and multiplication by a scalar are continuous functions
C.K/
C.K/ ! C.K/, R
C.K/ ! C.K/, the closure of a unital subalgebra
is a unital subalgebra. Therefore, the statement of the theorem will follow if we can
prove that every closed unital subalgebra of C.K/ which separates points is equal
to C.K/.
6.5
An important step in the proof of the theorem is the fact that the square root (and
hence the absolute value) of a non-negative continuous function on a bounded
compact interval
p is a uniform limit of polynomials. To prove this, we use the Taylor
expansion of 1 x.
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232 9 Metric and Topological Spaces II
p
Lemma. Let 0 < b < 1. Then the Taylor expansion of 1 x at the point x D 0
converges absolutely uniformly in the interval hb; bi.
While it is possible to prove this fact in an elementary way, a much easier proof
will follow from the methods of complex analysis. Because of this, we will skip
the proof at this point, and referpthe reader to Exercise (8) of Chapter 10 where we
define rigorously the function 1 x for x 2 C, Re.x/ < 1, and prove that the
(complex) radius of convergence of its Taylor series is 1.
Comment: In fact, using a lemma of Abel’s, the upper bound of uniform conver-
gence can be extended to 1. However, we do not need that fact.
1
!
p X 1=2
f C 1=n D .1 1=n f /k
k
kD0
1 1
max.f; g/ D .f C g C jf gj/; min.f; g/ D .f C g jf gj/: t
u
2 2
Let A C.K/ be a closed unital subalgebra which separates points, and let f 2
C.K/. Given " > 0, we will construct a g 2 A such that for every x 2 K,
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6 The space of continuous functions revisited: The Arzelà-Ascoli Theorem and : : : 233
Since " > 0 was arbitrary, this will imply that f is a limit of a uniformly convergent
sequence of elements of A, and hence f 2 A since A is closed. Since f was
arbitrary, A D C.K/, which implies the statement of the theorem.
To construct g, consider two points s ¤ t 2 K. Since A separates points, we
may choose h 2 A such that h.s/ ¤ h.t/. Now define, for v 2 K,
h.v/ h.t/
fs;t .v/ D f .s/ C .f .t/ f .s//
h.s/ h.t/
Then
and since fs;t ; f are continuous, Ut is open. On the other hand, s; t 2 Ut , and hence
.Ut /t ¤s is an open cover of K. Since K is compact, this open cover has a finite
subcover .Ut1 ; : : : ; Utm /. Putting
hs D min.fs;t1 ; : : : ; fs;tm /;
we have
Now let
Then
g D max.hs1 ; : : : ; hsp /:
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234 9 Metric and Topological Spaces II
Then g 2 A, and
as desired. t
u
7 Exercises
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7 Exercises 235
(10) Prove the following result known as the Weierstrass Approximation Theorem:
For a continuous function f W ha; bi ! R, there exists a sequence of
polynomials (with real coefficients) pn .x/ which, when restricted to ha; bi,
converge to f .
(11) Prove that the set of all polynomials in the variables sin.nx/, cos.nx/, n D
0; 1; 2; : : : is dense in C.h0; 2 i/. Is the set of all polynomials in the variables
sin.nx/, n D 0; 1; 2; : : : dense in C.h0; 2 /? Prove or disprove.
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In this chapter, we will develop the basic principles of the analysis of complex
functions of one complex variable. As we will see, using the results of Chapter 8,
these developments come almost for free. Yet, the results are of great significance.
On the one hand, complex analysis gives a perfect computation of the convergence
of a Taylor expansion, which is of use even if we are looking at functions of
one real variable (for example, power functions with a real power). On the other
hand, the very rigid, almost “algebraic”, behavior of holomorphic functions is a
striking mathematical phenomenon important for the understanding of areas of
higher mathematics such as algebraic geometry ([8]). In this chapter, the reader
will also see a proof of the Fundamental Theorem of Algebra and, in Exercise (4), a
version of the famous Jordan Theorem on simple curves in the plane.
1.1
Further recall from 4.2 of Chapter 8 that the set of complex numbers C is identified
with the Euclidean plane, with the distance jz1 z2 j equal to Euclidean distance
in R2 .
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238 10 Complex Analysis I: Basic Concepts
1.2
f .z C h/ f .z/
lim ;
h!0 h
(but this time in the metric space C), if it exists. If the limit exists, we speak (again)
of a derivative of f in z. More generally, one can introduce, in the obvious way,
partial derivatives of functions f W U1
Un ! C of several complex variables.
One uses the same notation as in the real case:
df
f 0 ; f 0 .z/; ; etc.
dz
By precisely the same procedure as in the real case we can prove the formulas
and the formula .zn /0 D n zn1 , so we can take derivatives of polynomials exactly
as in the real case.
1.3
f .z C h/ f .z/ zChz h
D D ;
h h h
an expression that has no limit for h approaching 0: on the real axis, i.e. for h D
h1 C i 0, we have constantly the value hh D hh11 D 1 while on the imaginary axis, i.e.
h2
for h D 0 C ih2 , we have h
h
D h2
D 1.
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1 The derivative of a complex function. Cauchy-Riemann conditions 239
In other words, while the condition of existence of complex derivative does imply
the existence of total differential of the function f considered as a map R2 ! R2
(or U ! R2 where U is an open set in R2 ), the converse is not true: the existence
of a complex derivative is a much stronger condition. We will see below in 5.3
that it has a different interpretation, namely of f preserving orientation and angles:
smoothness follows.
where P; Q are real functions in two real variables. We will now show that the
differentiability of f implies certain equations between the partial derivatives of
P an Q.
Remark. The equations (CR) are referred to as the Cauchy - Riemann conditions.
We have shown that these conditions are necessary for complex differentiability.
We will show in Theorem 1.5 below that the conditions are also sufficient when
f is continuously differentiable. A theorem of Looman and Menchoff states,
more generally, that the conditions are also sufficient assuming only that f is
continuous, but we will not need that result here. The conditions (CR) alone,
without any additional assumption on f , however, do not imply differentiability
(see Exercise (2).)
1 1
.f .z C h/ f .z// D .P .x C h1 ; y C h2 / P .x; y//
h h1 C ih2
(*)
i
C .Q.x C h1 ; y C h2 / Q.x; y//:
h1 C ih2
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240 10 Complex Analysis I: Basic Concepts
@P .x; y/ @Q.x; y/
Ci .D f 0 .z//
@x @x
and similarly (***) yields
@P .x; y/ @Q.x; y/
i .D f 0 .z//:
@y @y
Comparing the real and the imaginary parts, we obtain the desired equations. t
u
1.5 Theorem. Let P; Q be real functions of two variables with continuous partial
derivatives, let f .z/ D P .x; y/ C iQ.x; y/ and let the conditions (CR) be satisfied
at some point z D x C iy 2 U . Then f has a derivative in z.
Proof. We have
1
.f .z C h/ f .z/
h
1
D .P .x C h1 ; y C h2 / P .x; y/ C iQ.x C h1 ; y C h2 / iQ.x; y//
h
1
D .P .x C h1 ; y C h2 / P .x C h1 ; y/ C P .x C h1 ; y/ P .x; y/
h
C i.Q.x C h1 ; y C h2 / Q.x C h1 ; y/ C Q.x C h1 ; y/ Q.x; y///:
Denote the right-hand side by u. Using the Mean Value Theorem and (CR), we
obtain
P .x C h1 ; y C h2 / P .x C h1 ; y/ C P .x C h1 ; y/ P .x; y/
@P .x C h1 ; y C ˛h2 / @P .x C ˇh1 ; y/
D h2 C h1
@y @x
@P .x C h1 ; y C ˛h2 / @P .x C ˇh1 ; y/
D h2 C h1
@x @x
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1 The derivative of a complex function. Cauchy-Riemann conditions 241
and similarly
1.7
Recall the complex line integral from Section 4 above. Later we will need the
following fact. It is an easy consequence of 3.7 and 4.4 of Chapter 8, but we shall
spell things out, mainly to exercise the Cauchy-Riemann conditions.
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242 10 Complex Analysis I: Basic Concepts
F . / D P.˛; ˇ/ C i Q.˛; ˇ/
where
Z
P.˛; ˇ/ D .II/ .P .˛; ˇ; x; y/ Q.˛; ˇ; x; y/
Z
D .P .˛; ˇ; x; y/dx Q.˛; ˇ; x; y/dy/;
Z
Q.˛; ˇ/ D .II/ .Q.˛; ˇ; x; y/; P .˛; ˇ; x; y/
Z
D .Q.˛; ˇ; x; y/dx C P .˛; ˇ; x; y/dy/:
@P @Q @P @Q
D and D
@˛ @ˇ @ˇ @˛
@f @P @Q
so that F . / is holomorphic and hence has a derivative. By 1.4, D Ci
@ @˛ @˛
and hence by (1.7.1) and 1.4 again,
Z Z Z
@f .; z/ @P @Q @Q @P @P @Q dF
dz D .II/ ; C i.II/ ; D Ci D : t
u
@ @˛ @˛ @˛ @˛ @˛ @˛ d
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2 From the complex line integral to primitive functions 243
Proof. Put again f .z/ D P .x; y/ C iQ.x; y/. By 4.4 of Chapter 8, we have
Z Z Z
f D .II/ .P; Q/ C i.II/ .Q; P /
Li Li Li
and by the Green’s formula (5.4.1) of Chapter 8, the sum of these factors is equal to
Z Z
@Q @P @P @Q
Ci :
U @x @y U @x @y
2.2
2.3
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244 10 Complex Analysis I: Basic Concepts
In effect, this is trivial when the points a; u and u C h are colinear. Otherwise the
piecewise continuously differentiable simple curves P1 D L.a; u C h/ and P2 D
L.a; u/ C L.u; u C h/, h 2 h0; 1i, satisfy the assumptions of 2.2 and hence (2.3.1)
follows from 3.4 and 4.4 of Chapter 8. Now, by (2.3.1),
Z Z 1
1 1 1
.F .u C h/ F .u// D f .u C th/hdt D f .u C th/dt
h h 0 0
Z 1 Z 1
D P .u C th/dt C i Q.u C th/dt
0 0
which with real h ! 0 approaches P .u/ C iQ.u/, by the Mean Value Theorem. u
t
2.4 Comment
2.5
It is curious to observe that the proof of Theorem 2.3 can be “transported” (with only
minor modifications) by a (real) injective regular map. More precisely, identifying
C with R2 , let W U ! V be a bijective regular map in the sense of Subsection 7.1
of Chapter 3. Then the proof of Theorem 2.3 remains valid with the line segments
L.a; b/ replaced by their -images. We obtain therefore the following
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3 Cauchy’s formula 245
Proposition. If V is an open set in C such that there exists a bijective (real) regular
map W U ! V where U is convex, then every holomorphic function on V has a
primitive function.
As it turns out, the converse is also true. In fact, in Section 1 of Chapter 13, we
shall prove much more, namely that unless U D C, the map can be chosen to be
holomorphic. This is the famous Riemann Mapping Theorem.
3 Cauchy’s formula
3.1 Lemma. Let Kr be a circle with center in a point z and radius r > 0, oriented
counter-clockwise. Then we have
Z
d
D 2 i:
Kr z
Proof. Parametrize Kr by
3.2
Notice that the integral computed in 3.1 is not required to vanish by Theorem 2.1
because the argument is not defined (and in fact, goes to infinity) at D z.
Proof. We have
Z
f ./
d
Kr z
Z Z
f .z/ f ./ f .z/
D d C d:
Kr z Kr z
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246 10 Complex Analysis I: Basic Concepts
The first summand on the right-hand side is equal to 2 if .z/ by 3.1. We shall prove
that the second summand is 0. Since
f ./ f .z/
f 0 .z/ D lim ;
!z z
f ./ f .z/
the quantity is bounded on the set U Xfzg for some open neighborhood
z
U of z (and hence, by continuity, on
.z; r/ X fzg). Let
ˇ ˇ
ˇ f ./ f .z/ ˇ
ˇ ˇ < A in
.z; r/ X fzg:
ˇ z ˇ
In particular,
Z
f ./ f .z/
lim d D 0:
s!0 Ks z
U D
.z; r/ X
.z; s/; (*)
D lim 0 D 0: t
u
s!0
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3 Cauchy’s formula 247
Z
kŠ f ./
f .k/ .z/ D d: (3.4.1)
2 i Kr . z/kC1
t
u
Proof. Using Cauchy’s formula (Theorem 3.3) with f replaced by fn , and taking
the limit after the integral sign using Lebesgue’s Dominated Convergence Theorem
implies the same formula for f , proving that f is holomorphic. Further, using the
same argument on formula (3.4.1) (k D 1), we see that fn0 converges to f 0 , and
further that the convergence is uniform in a disk with center z and radius r=2. A
compact set is covered by finitely many such disks by the Heine-Borel Theorem 2.3
of Chapter 9, which implies that the convergence of derivatives is uniform on a
compact set. t
u
The following result will be useful for applying the Arzelà-Ascoli Theorem 6.2
to sequences of analytic functions.
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248 10 Complex Analysis I: Basic Concepts
4C jz z0 j
: (C)
r
Now let K U be a compact subset. We claim that there exists an r > 0 and a
compact set L, K L U such that every point of distance r from some point
of K belongs to L.
(For every point x 2 K, there is a number s.x/ > 0 such that
.x; s.x// U .
By the Heine-Borel Theorem 2.3 of Chapter 9, K is covered by finitely many of the
open disks
.xi ; s.xi /=3/, for some points xi , i D 1; : : : ; k. Let s D minfs.xi /ji D
[k
1; : : : ; kg. Then we may put r D s=3, L D
.xi ; s.xi /=3/.
i D1
Now let C be a uniform bound on jfn .z/j for z 2 L. Then in (C) we may always
use these values of C and r. We see that then at least for z; t 2 K, jz tj < r=2,
4C jz tj
jfn .z/ fn .t/j < ;
r
which implies equicontinuity on K. t
u
Note that in the preceding proof, we have proved more than equicontinuity,
namely a uniform Lipschitz constant.
3.8 Remarks
1. Note that the statements 3.4 and 3.5 are in sharp contrast with real analysis.
2. We will see that Cauchy’s formula in complex analysis plays an analogous role
to the Mean Value Theorem in real analysis. It is, however, a much stronger tool,
which makes certain concepts (such as the Taylor series) much easier.
f ./
3. Realize the role of the argument going to infinity at the point z. Note that
z
all the information about the integral in 8.3 is contained in an arbitrarily small
neighborhood of z.
4. By the same argument, the circle Kr could be replaced by any closed simple
curve L which is the boundary of a domain U oriented counter-clockwise and
such that z 2 U .
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4 Taylor’s formula, power series, and a uniqueness theorem 249
1 0 1 1
f .z/ D f .c/C f .c/.zc/C f 00 .c/.zc/2 C C f .n/ .c/.zc/n C: : : :
1Š 2Š nŠ
Proof. We have
1 z 1
D : (*)
z c 1 zc
c
1 1 1
D C .z c/ C .z c/2 C :::
c . c/2 . c/3
1
C.z c/n C ::::
. c/nC1
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250 10 Complex Analysis I: Basic Concepts
4.2
Note that repeating verbatim the proofs in Section 7 of Chapter 1, we get the
following
converges absolutely and uniformly in a circle with center c and any radius
1
s < r D lim inf p
n
jan j
and diverges outside of the closed circle with center c and radius r. (The number r
is called the radius of convergence of the power series (*).)
Moreover, the power series
1
X
kak .z c/k1
kD1
has the same radius of convergence as (*), and the series (*) may be differentiated
term by term.
4.3
z z2
ez D 1 C C C :::;
1Š 2Š
z3 z5
sin.z/ D z C :::;
3Š 5Š
z2 z4
cos.z/ D 1 C :::
2Š 4Š
will now be considered the definitions of the functions e z , sin.z/, cos.z/ for z
complex (the radius of convergence of these series is 1). Therefore, we have
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4 Taylor’s formula, power series, and a uniqueness theorem 251
and also
e iz C e iz e iz e iz
cos.z/ D ; sin.z/ D :
2 2i
Assuming (*) does not hold, let n be the smallest number such that an ¤ 0. Then in
some neighborhood of c,
The function in the parentheses on the right-hand side is continuous (it is a uniform
limit of continuous functions), and not zero at c; thus, it is non-zero in some
neighborhood of c, and so is .z c/n , contradicting our assumptions. t
u
Proof. Let
M is clearly open, and by the lemma, it is also closed and non-empty. Since U is
connected, we have M D U . t
u
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252 10 Complex Analysis I: Basic Concepts
Note that on two power series of the form 4.2.(*) with the same c, we can perform
addition, sutraction and multiplication (in the case of multiplication, note that only
finitely many terms with the same power .zc/k are added). As an inverse operation
to this purely algebraic multiplication, note that it is also possible to divide by any
power series 4.2.(*) with a0 ¤ 0, figuring the coefficients of the ratio by a recursive
procedure.
It will be important for us that when these purely algebraic operations are
performed on power series with a positive radius of convergence representing Taylor
series at c of holomorphic functions f , g, the power series resulting in an algebraic
operation converges and is the Taylor series of f C g, f g, f g or f =g, (the
division requires g.c/ ¤ 0). All of these statements are more or less obvious with
the exception of the division. Here we note that since g.c/ ¤ 0, we have g.z/ ¤ 0 in
some disk
.c; r/, r > 0. Therefore, f =g is a holomorphic function in a disk with
center a, and hence has a Taylor expansion at c. Multiplying this Taylor expansion
with the Taylor expansion of g at c algebraically, we then get the Taylor expansion
of f at c by uniqueness. This implies that the Taylor expansion of f =g at c is the
algebraic ratio of the Taylor expansions of f and g at c.
Proof. By the formula from 3.4, for any circle Kr with center z and radius r we have
Z
0 2Š f ./
f .z/ D d:
2 i Kr . z/2
2Š A 8A
jf 0 .z/j 4 2 r 2 D :
2 r r
Since r > 0 was arbitrary, we must have f 0 .z/ 0 and hence f must be constant.
t
u
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5 Applications: Liouville’s Theorem, the Fundamental Theorem of Algebra: : : 253
R D 2n max.ja0 j; : : : ; jan j/
On the other hand, on fzj jzj Rg, f is bounded because it is continuous. Thus, f
is bounded on all of C, and by Liouville’s Theorem, it is constant, and hence so is
p.z/. This is a contradiction since we assumed n 1. t
u
5.3
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254 10 Complex Analysis I: Basic Concepts
Proof: This is really a statement entirely about the R-linear map Dfz for each
z 2 U (see Exercise (1)), which is a consequence of the following
for some R1 < s < jz cj. The exact choice of r or s does not change the value by
Theorem 2.1. Furthermore, we have
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6 Laurent series, isolated singularities and the Residue Theorem 255
To see this, consider a circle K with center z and a small radius oriented
counterclockwise, and apply Theorem 2.1 to the function
f ./
z
of the variable with simple closed curves Lr , Ls , K , along with Cauchy’s
formula (Theorem 3.3).
By differentiating under the integral sign (Theorem 1.7), and the fact that the
value does not depend on r, we see that the function f1 .z/ is holomorphic in the
disk jz cj < R2 , and hence has a Taylor expansion. In case of the function f2 .z/,
it is convernient to perform the substitution
1 1 1
D ; D c C ; d D 2 d;
c
and similarly
1 1
tD ; zDcC ;
zc t
so that
1 t
D :
z t
where M is the circle with center 0 and radius 1=s < 1=jtj oriented counterclock-
wise (note that the substitution reverses orientation, so we have a total of 4 minus
signs, which result in a plus). Again by differentiating under the integral sign, we
see that g.t/=t is a holomorphic function in the circle jtj < 1=R1 , and hence has
a Taylor expansion. (Note: when performing the substitution, we implicitly used
the fact that when performing substitution in complex line integrals, we may treat
differentials the same way as in ordinary single-variable integral substitution - see
Exercise (11) below). Writing the Taylor series of g.t/ in the variable .z c/, we
obtain an expansion of the form
X
f2 .z c/ D an .z c/n ;
n<0
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256 10 Complex Analysis I: Basic Concepts
Proof. The existence of the expansion (6.1.2) follows from the expansions for the
functions f1 , f2 in the variable z c discussed above. Moreover, the convergence
properties of the series (6.1.2) follow from our already discussed theory of power
series. Regarding uniqueness, note that the coefficients an can be calculated by
Cauchy integrals, which can be performed term by term by the convergence
properties of the power series (see Exercise (13) below). t
u
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6 Laurent series, isolated singularities and the Residue Theorem 257
Proof. Assuming the opposite, there exists an A 2 C and an " > 0 and a ı > 0 such
that f .z/ A ¤ 0 for z 2
.c; ı/. But then the function
1
f .z/ A
has a removable sinularity at A, and hence f .z/ has at most a pole at A. t
u
Since we may integrate the Laurent series term by term, it follows that if L is a circle
in U with center c oriented counter-clockwise such that the interior of the circle is
also contained in U , then
Z
1
reszDc f .z/ D f ./d: (6.2.2)
2 i L
From this and Theorem 2.1, we then immediately get the following fact:
t
u
The Residue Theorem has the following celebrated consequence. We say that a
function is meromorphic in an open set U C if f is holomorphic and non-zero
on U X S for a discrete set S U , and f has at most a pole at each c 2 S . Then
we define the degree of f at c 2 U as
8
< n if f has a zero of degree n at c
degc .f / D n if f has a pole of order n at c
:
0 otherwise.
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258 10 Complex Analysis I: Basic Concepts
X k Z X
1 f 0 .z/
dz D degc .f /:
j D1
2 i Li f .z/ c2U
(Note that since U is compact, the sum on the right-hand side has only finitely many
non-zero terms.)
so that
f 0 .z/ n g 0 .z/
D C ;
f .z/ zc g.z/
and hence
f 0 .z/
reszDc D n:
f .z/
The statement then follows directly from the Residue Theorem (see Exercise (17)).
t
u
e Ln.z/ D z;
we have
If U C is, say, a convex open set on which f .z/ has no zero, then f 0 .z/=f .z/
has a primitive function Ln.f .z// whose imaginary part differs from Arg.f .z// by
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6 Laurent series, isolated singularities and the Residue Theorem 259
2 k, k 2 Z. The whole point is, however, that by Lemma 3.1, Ln.z/ cannot be well-
defined on the whole set C X f0g; roughly speaking, when we follow a circle with
center 0 once around counter-clockwise, the value of the logarithm will increase by
2 i (note that its real part won’t change: it is just its imaginary part, the argument,
which will inrease by 2 ). Thus, Theorem 6.3.1 in the case k D 1 makes precise the
intuitive assertion that following around a simple closed curve on which f .z/ has
no zero and which is a boundary oriented counter-clockwise of a domain U , then
the increase of the argument of f along this curve is equal to 2 times the number
of zeros of f inside U .
Let f be a holomorphic function on U which is non-zero outside of a finite set of
points. Then f is meromorphic, and the sum of degrees of f at all the points a 2 U
(which has only finitely many non-zero summands) is called the number of zeros of
the function f in the set U . (Thus, this is a count of zeros with “multiplicities”.)
Then f , g have the same number of zeros in U . (Note that again, since U is
compact, by Theorem 4.4 ,f and g have only finitely many zeros in U .)
F ŒL1 q q Lk
where
is is the open disk with center 1 and radius 1. Then 1=z has a primitive
function on
, which we will denote by Ln.z/. The chain rule then implies
F 0 .z/
.Ln.F .z///0 D :
F .z/
Therefore,
X k Z
1 F 0 .z/
dz D 0;
j D1
2 i Li F .z/
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260 10 Complex Analysis I: Basic Concepts
.z0 ; "/ is n.
Proof. Let L" be the circle with center z0 and radius " oriented conterclockwise. We
will study the integral
Z
1 f 0 .z/
dz: (*)
2 i L" f .z/ c
Proof. Note that in particular in the conclusion of Theorem 6.3.3, every element of
Proof. By Corollary 6.3.4, for any z 2 U , all points in a neighborhood of f .z/ are
in the image of f , so this will include points of greater absolute value. t
u
Proof. We know from Weirstrass’s Theorem (Theorem 3.6) that f .z/ is a holomor-
phic function on U . Suppose f .z/ is not identically 0. Then by Theorem 4.4, for
any point z0 2 U , there exists a number r > 0 such that f .z/ is defined and not
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6 Laurent series, isolated singularities and the Residue Theorem 261
Now by the Argument Principle, the argument of the limit in (*) is the number of
zeros of fn inside the circle K, which is 0, while the right-hand side is the number
of zeros of f inside K. In particular, f .z0 / ¤ 0, and the statement follows, since z0
was arbitrary. t
u
X1
1
.s/ D s
for Re.s/ > 1:
mD1
m
A lot can be said about the Riemann zeta function, but here we want to show how
the Residue Theorem can be applied to evaluating .k/ for k 2 an even integer,
which is a typical example of an application of the theorem. (The evaluation of .k/
for odd integers k > 2 is still an open problem.) First, note that e z 1 has a simple
(=order 1) zero at z D 0, and hence
z
ez 1
X Bj 1
z
D zj :
ez 1 j D0
j Š
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262 10 Complex Analysis I: Basic Concepts
2 i
f .z/ D :
zk .e 2 i z 1/
.2 i /k Bk
reszD0 f .z/ D :
kŠ
On the other hand, clearly f .z/ has a simple (D order 1) pole at m 2 Z X f0g), and
using Taylor series at z D m, one gets
1
reszDm f .z/ D :
mk
Also, clearly, f .z/ has no other poles. Let L be a rectangle with sides
˙.n C 21 / C ti, niCt, t 2 R in the appropriate ranges, oriented counterclockwise.
By the Residue Theorem,
Z !
.2 i /k Bk Xn
1
f .z/dz D 2 i C2 : (C)
L kŠ mD1
mk
On the other hand, the left-hand side tends to 0 with n ! 1. In effect, we claim that
je 2 iz 1j > C (*)
.2 i /k Bk
.k/ D :
2.kŠ/
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7 Exercises 263
7 Exercises
(1) Prove from first principles that for a holomorphic function f W U ! C where
U is open, f , thought of as a map from an open set of R2 to R2 , has a total
differential at every point.
(2) Prove that the function of one complex variable
(
4
e z if z ¤ 0
f .z/ D
0 if z D 0
x W Vx !
.0; 1/
with
det.Dx / > 0;
c.b/ D x;
and
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264 10 Complex Analysis I: Basic Concepts
indc .z/ D 0:
5. Prove that there exists a point x of cŒha; bi for which, in the notation of part
2 of this Exercise, indc .z1 / D 0 or indc .z2 / D 0. Note that either alternative
can arise depending on the orientation of c. [Hint: Let x 2 I m.c/ be a point
with maximal real part.]
6. Let Ui be the connected component of C X cŒha; bi which contains the
point zi , i D 1; 2. Prove that Ui X Ui D cŒha; bi. [Hint: Use part 1 and
compactness.]
7. Prove from part 5 that U1 [ U2 [ cŒha; bi is open, and equal to its
closure, hence equal to C. Hence, CXcŒha; bi has precisely two connected
components, namely U1 and U2 (note that, by parts 2 and 3, U1 ¤ U2 ).
(5) Prove that the set of all z 2 C such that e z D 1 is precisely the set
f2k i j k 2 Zg. [Hint: Recall Exercises (12), (11) of Chapter 1].
(6) Prove that if Re.t/ > 0, then there exists a unique z 2 C with =2 <
Im.z/ < =2 such that e z D t. Denote z D ln.t/. Prove that the complex
derivative of ln.z/ is 1=z.
(7) For Im.z/ > 0, a 2 C, define za D e a ln.z/ . Mimic Exercise (7) of Chapter 1 to
show that the complex derivative of za is aza1 .
(8) Define, for a 2 C,
!
a
D1
0
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7 Exercises 265
Prove Newton’s formula, which states that for z 2 C with jzj < 1, we have
1
!
X a
.1 C z/a D zn :
nD0
n
(9) Suppose that f is a holomorphic function on C, and suppose there exist non-
zero numbers a; b 2 C such that we do not have qa D b for any q 2 Q, and
such that f .z C a/ D f .z/, f .z C b/ D f .z/ for all z 2 C. Prove that then f
is constant. [Note that there is more than one case to consider.]
(10) Prove that a non-constant holomorphic function on f W C ! C satisfies
f ŒC D C. [Hint: If a … f ŒC, then the function 1=.f .z/ a/ is holomorphic
and bounded.]
(11) Prove that if L is a parametrized oriented piecewise smooth curve in an open
set U C, h W U ! C is a holomorphic injective function and f is a
holomorphic function on hŒU , then
Z Z
f .t/dt D f .h.z//h0 .z/dz:
hŒL L
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266 10 Complex Analysis I: Basic Concepts
(15) Prove that the complex function f .z/ D e 1=z for z ¤ 0 has an essential
singularity at z D 0. Conclude that the Taylor expansion of f at a 2 C X f0g
has radius of convergence jjajj. (Compare to Exercise (13) of Chapter 1.)
(16) Prove that a function f as in Subsection 6.2 has a pole of order n at a if and
only if g.z/ D f .z/.z a/n is holomorphic in U and g.a/ ¤ 0. Similarly,
prove that f has a zero of order n at a if and only if h.z/ D f .z/=.z a/n is
holomorphic in U and h.z/ ¤ 0.
(17) Let U be a connected open subset of C and let f; g be meromorphic functions
on U . Prove that f g, f =g are meromorphic on U .
(18) Prove that Bk D 0 if k > 2 is an odd integer.
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Multilinear Algebra
11
1.1
In this Chapter, the symbol F stands for either the field R of real numbers or the
field C of complex numbers. Let V , W be vector spaces over F. Denote by
HomF .V; W /
f W V ! W:
Observe that HomF .V; W / is again a vector space: for f; g 2 HomF .V; W /, we have
a linear map f C g 2 HomF .V; W / defined by
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268 11 Multilinear Algebra
and call V the dual of the vector space V and often refer to elements of V as
linear forms on V .
by composition with :
The map is clearly linear. Moreover, this construction clearly preserves the
identity, and if we have another linear map W W 0 ! W 00 , we have
ı D . ı / :
g 7! .g/ D g ı :
This map,
however, goes in the opposite direction! Again, is clearly linear, and this
construction preserves identity. Also, it preserves composition, but this time in the
reversed order: If W V 0 ! V 00 is a linear map, then
. ı / D ı :
This behavior, i.e. reversing the direction of maps and the order of composition, is
referred to as contravariance and the opposite of contravariance, i.e. preserving the
direction of maps and order of composition, is then referred to as covariance. Thus,
the construction is contravariant and the construction is covariant.
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1 Hom and dual vector spaces 269
WV !W
W W ! V :
1 when i D j
fi .vj / D
0 else.
We conclude that
Note that finite-dimensionality was used in the last line of the proof, where we
would get an undefined infinite sum, were the basis infinite.
W V ! .V /
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270 11 Multilinear Algebra
v2V
and
.f W V ! F/ 2 V :
Then define
..v//.f / D f .v/:
Proof. Let V have an ordered basis .v1 ; : : : ; vn /. Let .f1 ; : : : ; fn / be the dual
ordered basis, and let .w1 ; : : : ; wn / be the dual ordered basis of .f1 ; : : : ; fn /. By
definition, we have .vi / D wi . t
u
WV !V (*)
by
V ! V :
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2 Multilinear maps and the tensor product 271
2.1
and
V1
Vn
multi
multi
W0 W:
0
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272 11 Multilinear Algebra
The dotted arrow means that the map (in this case linear) exists and is determined
by the other data. For given vector spaces V1 ; : : : ; Vn , such a universal vector space
W0 indeed exists. It is called the tensor product, and denoted by
W 0 D V1 ˝ ˝ Vn :
Of course, the existence is yet to be proved. However, let us observe that just from
the universal property, if the tensor product exists, it must be unique up to a preferred
(we say canonical) isomorphism: Suppose
0 W V1
Vn ! W00
0 D :
D 0 :
D ;
D Id;
and similarly
D Id;
Proof. The construction is not very inspiring. Recall from Appendix A, 5.6 the
construction of the free vector space. Now take the free vector space
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2 Multilinear maps and the tensor product 273
F.V1
Vn / (*)
0 W V1
Vn ! F.V1
Vn /;
x D .v1 ; : : : ; vn / 2 V1
Vn
to the free generator of the same name. This map 0 is just a map of sets; there is no
reason even to suspect that it may be multilinear.
Now, however, we apply our technique of factorization. Namely, in (*), take the
vector subspace Z generated by all the elements
and
W0 D F.V1
Vn /=Z:
W F.V1
Vn / ! W0 :
Put
D 0 :
W V1
Vn ! W
ˆ W F.V1
Vn / ! W
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274 11 Multilinear Algebra
such that
ˆ.v1 ; : : : ; vn / D .v1 ; : : : ; vn /:
ˆŒZ D 0;
0 W F.V1
Vn /=Z ! W
such that
0 D ˆ: t
u
v1 ˝ ˝ vn D .v1 ; : : : ; vn / 2 V1 ˝ ˝ Vn :
Let us also remark that to be completely precise, we should denote our tensor
product by
V1 ˝F ˝F Vn
to distinguish the field. We will, however, typically not use this longer notation
unless confusion can arise.
Perhaps the most important convention is that in most of advanced mathematics,
a multilinear map
W V1
Vn ! W
0 W V1 ˝ ˝ Vn ! W;
which means that the two concepts are no longer distinguished explicitly, and the
linear variant is written in all formulas.
To avoid excessive indexing, assume here that n D 2 and investigate the tensor
product V ˝ W of vector spaces V , W with ordered bases .v1 ; : : : ; vm / and
.w1 ; : : : ; wp /. (See Exercise (2).)
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2 Multilinear maps and the tensor product 275
fvi ˝ wj j i D 1; : : : ; m; j D 1; : : : ; pg
is a basis of V ˝ W .
0 W V
W ! F.V
W /
X
m X
n X
0 . i vi ; j wj / D i j vi ˝ wj :
i D1 j D1 i;j
WV
W !U
X
m X
n X
. i vi ; j wj / D i j .vi ; wj /;
i D1 j D1 i;j
so the map required by the universality property is uniquely given by the formula
0 .vi ˝ wj / D .vi ; wj /: t
u
W V ˝ W ! Hom.V; W /;
W V ˝ W ! .V ˝ W / :
.f W V ! F/ 2 V ;
.g W W ! F/ 2 W :
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276 11 Multilinear Algebra
Define
Proof. Let V; W have ordered bases .v1 ; : : : ; vm / and .w1 ; : : : ; wn /, let the dual
ordered bases be .f1 ; : : : ; fm /, .g1 ; : : : ; gn /. We already know that the space space
Hom.V; W / is isomorphic to the space of .m
n/-matrices by assigning to a linear
map W V ! W its matrix with respect to the bases .vi /, .wj /. Denote by
i;j 2 Hom.V; W /
the linear map whose matrix has 1 in the j ’th row and i ’th column and 0’s elsewhere.
Then clearly the set of all i;j , i D 1; : : : ; m; j D 1; : : : ; n is a basis of Hom.V; W /,
and we have
.fi ˝ wj / D i;j ;
.fi ˝ gj / D ei;j ;
Let V , W be vector spaces over the field F (which, again, we assume to be equal to
R or C). Recall that a multilinear map
W„
V
ƒ‚
V… ! W
k times
W V ˝˝V ! W
„ ƒ‚ …
k times
(the left hand side is, of course, also denoted by V ˝k ). The multilinear map is
called alternating if for any permutation
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3 The exterior (Grassmann) algebra 277
3.2
It is natural to ask if there is a universal object for alternating multilinear maps just
as the tensor product was for multilinear maps, i.e. if for every vector space V and
every k D 0; 1; 2; : : : there exists a vector space Wa and an alternating map
W V ˝k ! Wa
W V ˝k ! W
D a ;
V ˝k
alt
alt
Wa W:
a
The notation alt means an alternating map. Such an object indeed exists, as we shall
prove in 3.3. It is called the exterior power, and is denoted by ƒk .V /. It is also
unique up to canonical isomorphism by the same argument as the tensor product
(see Exercise (6)).
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278 11 Multilinear Algebra
Proof. Let Z V ˝k be the vector subspace generated by all elements of the form
W V ˝k ! ƒk .V /
ŒZ D 0;
a W ƒk .V / D V ˝k =Z ! W
v1 ^ ^ vk D .v1 ˝ ˝ vk / 2 ƒk .V /:
is a basis of ƒk .V /.
Proof. Again, we will use the uniqueness which follows from the universal property.
Let Wa0 be the free vector space on the set (1). A linear map on a vector space can
be defined by specifying its values on the basis elements. The basis elements on
V ˝k are
Define thus
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3 The exterior (Grassmann) algebra 279
0 W V ˝k ! Wa0
by sending the element (2) to 0 if two of the numbers i1 ; : : : ; ik are equal, and to
if
a0 W Wa0 ! W
by
follows from the definition of an alternating map (in particular, note that if two
coordinates of x coincide, swapping these two coordinates only changes the sign but
not x so we get .x/ D .x/, implying .x/ D 0). Note also that the definition
(3) is thereby forced by (4), so 0 has the universal property of 3.2. t
u
3.5 Remark
dim.ƒn .V // D 1:
W V ! Fn ;
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280 11 Multilinear Algebra
(where the right hand side is the space of columns), one such non-zero alternating
map is
W V ˝k ˝ Z ˝` ! W
which are alternating in the first k coordinates and the last ` coordinates separately.
By this, we mean that
2 D ˝ W V ˝k ˝ Z ˝` ! ƒk .V / ˝ ƒ` .Z/
(see Exercises (3) and (4)) is alternating in the first k and last ` coordinates
separately. For any vector space W and any linear map
W V ˝k ˝ Z ˝` ! W
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3 The exterior (Grassmann) algebra 281
alternating in the first k and last ` coordinates separately, there exists a unique
linear map
2 W ƒk .V / ˝ ƒ` .V / ! W
such that D 2 2 .
Proof. It is obvious that 2 is alternating in the first k and the last ` coordinates
separately. Consider a map as in the statement of the proposition. Then for
w 2 Z ˝` fixed,
v 7! .v ˝ w/
w W ƒk .V / ! W: (*)
Fixing now v 2 ƒk .V /, on the other hand, (*) is clearly linear and alternating in w,
thus giving us a map
v W ƒ` .V / ! W: (**)
ƒk .V /
ƒ` .Z/ ! W;
2 W ƒk .V / ˝ ƒ` .Z/ ! W:
The whole point of the proposition for our purposes is that for V D Z,
when a map
W V ˝kC` ! W
^ W ƒk .V / ˝ ƒ` .V / ! ƒkC` .V /:
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282 11 Multilinear Algebra
We will think of this map as a kind of a product, called the exterior product, i.e.
write, for x 2 ƒk .V /, y 2 ƒ` .V /,
x ^ y 2 ƒkC` .V /:
x ^ y D .1/k` y ^ x
since .1/k` is the sign of the permutation swapping the first k with the last `
coordinates (without changing their individual orders). Note that if we put
1
M
ƒ.V / D ƒk .V /
kD0
.x ^ y/ ^ z D x ^ .y ^ z/; 1 ^ x D x ^ 1 D x:
One calls ƒ.V / the exterior algebra (or the Grassmann algebra).
W ƒk .V / ! .ƒk .V //
by
X
..f1 ^ ^ fk //.v1 ^ ^ vk / D sgn./ f .1/ .v1 / f .k/ .vk /
where the sum is over all permutations on the set f1; : : : ; kg.
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3 The exterior (Grassmann) algebra 283
Proof. Let .v1 ; : : : ; vn / be an ordered basis of V and let .f1 ; : : : ; fn / be the dual
ordered basis of V . Then for 1 i1 < < ik n, we have
where the sum is over all permutations on f1; : : : ; kg. It is useful to note that
if .v1 ; : : : ; vn / is an ordered orthonormal basis of V , then the basis given by
Proposition 3.3 is orthonormal.
Now let V be an oriented real finite-dimensional vector space of dimension n.
Recall from Remark 3.5 that dim.ƒn .V // D 1 and note that an orientation specifies
a connected component C of ƒn .V / X f0g. Now there exists a unique 2 C with
h; i D 1. There exists a unique linear isomorphism
" W ƒn .V / ! R
such that
"./ D 1:
W ƒk .V / ! .ƒnk .V //
defined by
W ƒk .V / ! ƒnk .V /
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284 11 Multilinear Algebra
as the composition
ƒk .V / .ƒnk .V // ƒnk .V /
v1 ^ ^ vn D :
Then we see readily that for an oriented ordered basis .v1 ; : : : ; vn / of V , we have
.v1 ^ ^ vk / D vkC1 ^ ^ vn :
4 Exercises
V ˝ W ! W ˝ V;
V ˝ .W ˝ Z/ ! .V ˝ W / ˝ Z;
F˝V !V
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4 Exercises 285
Š
ƒk .V / .ƒk .V //
Š Š
ƒk .V / ƒk .V /
Id
where the vertical maps are given by the inner products in V and ƒk .V /.
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1 Smooth manifolds
hx W Ux ! Rn (*)
(open map means that the image of every set open in the domain is open in the
codomain). The neighborhood Ux is called a coordinate neighborhood, and the
function hx is called a coordinate system, or coordinate system at x. The map
assigning to each x 2 M a coordinate neighborhood and a coordinate system is
called an atlas. The coordinate systems of an atlas are also referred to as charts.
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288 12 Smooth Manifolds, Differential Forms and Stokes’ Theorem
Remarks:
1. Note that instead of requiring hx to be open, we could have equivalently required
that hx be a homeomorphism (see Exercise (1)).
2. Since we assume M is separable and metrizable, it has a countable basis (see 1.2
of Chapter 9). The reason we don’t actually say M is a metric space is that we do
not want to specify the metric: the metric has no geometric significance, and is
only a technical tool at this point. While there are metrics on manifolds which do
have a geometrical significance, we will only see these when we develop more
structure (such as the concept of a Riemann metric in Chapter 15).
3. Note that the pairs .Ux ; hx / may coincide for different points x. For example,
for M D Rn , the atlas may contain only one coordinate system, namely Rn with
the identity map Id W Rn ! Rn , which can be equal to .Ux ; hx / for all x 2 Rn .
In other interesting cases, the atlas may contain only finitely many coordinate
systems (in fact, note that by definition, a compact manifold always has such
a finite atlas). The reader may wonder why we don’t simply speak of atlases
as open covers U, with coordinate systems on each U 2 U. This is merely a
technical point: it turns out that being able to denote a coordinate neighborhood
of a point by a single symbol simplifies many arguments.
4. Because we required separability, by our definition, an uncountable discrete set is
not a manifold. There is an alternative definition, calling a manifold any (possibly
a
uncountable) disjoint union of manifolds in our sense. (In a disjoint union Mi ,
i
a set U is open if and only if each U \ Mi is open in Mi .)
.hx /1 hy
hx ŒUx \ Uy Ux \ Uy hy ŒUx \ Uy (C)
is a smooth map, i.e. a map which is continuous and has partial derivatives of all
orders which are also continuous. (Note that the domain and codomain of (C) are
open subsets of Rn ; also note that the intersection of Ux and Uy may be empty; in
that case, the condition (C) is void.)
Remarks:
1. Note that this definition is completely intuitive: it simply says that in a coordinate
neighborhood, we can speak of smooth real functions, and that these concepts are
compatible when we pass from one coordinate neighborhood to another.
2. Note that the continuity of all higher partial derivatives does not follow from
their existence, even on an open set (see Exercise (2) of Chapter 3).
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1 Smooth manifolds 289
h1
x f hf .x/
hx Œ.f 1 ŒUf .x/ / \ Ux f 1 ŒUf .x/ \ Ux Uf .x/ Rn
1.4 Examples
(1) Any open subset of a Euclidean space Rn is, of course, a smooth manifold,
and C r -maps between such manifolds are simply maps for which the required
partial derivatives (in the old sense) exist and are continuous.
(2) More generally, an open subset U of a smooth manifold M automatically
inherits a structure of a smooth manifold.
(3) Suppose f W Rn ! R is a C 1 -function. Define
.x; f .x// 7! x:
The smooth manifold M is known as the graph of the function f . The identity
embedding M RnC1 is a C 1 -map and the projection
M ! Rn
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290 12 Smooth Manifolds, Differential Forms and Stokes’ Theorem
given by
.x; f .x// 7! x
is a diffeomorphism.
(4) The first “non-trivial” example of a smooth manifold is the n-sphere
X
n
S n D f.x0 ; : : : ; xn / 2 RnC1 j xi2 D 1:g:
i D0
u1
i Œ.0; 1i Ui
(i.e. the support of ui is contained in Ui ), and for every x 2 M there exists an open
neighborhood Vx of x and a finite subset Ix I such that for all y 2 Vx , i 2 I XIx ,
we have ui .y/ D 0 and
X
ui D 1: (1.5.1)
i 2I
(Note that the expression on the left-hand side of (1.5.1) makes sense because on
Vx , it can be defined as the sum over Ix .)
A refinement of an open cover .Ui /i 2I is an open cover .Vj /j 2J such that for
every j 2 J , there exists an i 2 I such that Vj Ui . A cover .Ui /i 2I is called
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1 Smooth manifolds 291
locally finite if for every x 2 M , there exists an open neighborhood Vx and a finite
subset Ix I such that for i 2 I X Ix , Vx \ Ui D ;.
Lemma. For every open cover .Ai /i 2I of a smooth manifold M , there exists an
atlas .Uj ; hj /j 2J such that J is countable, the cover .Uj / is locally finite, is a
refinement of the cover .Ai /, we have hj ŒUj D
.o; 3/ and .h1j Œ
.o; 1//j 2J is
also a cover of M .
Proof. Since M has a countable basis by Theorem 1.2 of Chapter 9, any open
cover has a countable subcover. Since clearly every point of M has a compact
neighborhood, there exists a countable cover by open sets whose closures are
compact, which is a refinement of .Ai /. Assume, without loss of generality, that
.Ai /i 2I itself is such a cover, and that, moreover, I D f1; 2; : : : g. Now define
K1 D A1 , and assuming K1 ; : : : ; Ki are defined, let
Ki C1 D A1 [ [ Ar
where r > i is the smallest number such that Ki A1 [ [ Ar . (Note that such
a number exists by compactness.)
Denote by X ı the interior of a set M , i.e.
X ı D M X .M X X /:
and
Ki 1 Kiı :
Theorem. For any open cover .Ai / of a smooth manifold M there exists a smooth
partition of unity subordinate to .Ai /.
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292 12 Smooth Manifolds, Differential Forms and Stokes’ Theorem
.2 jjxjj/
g.x/ D
.2 jjxjj/ C .jjxjj 1/
Now take the atlas .Uj ; hj / from the statement of the Lemma, let gj D g ı hj and
define
gj
uj D X for j 2 J :
gk
k2J
The notion of a tangent vector to a smooth manifold models the geometric intuition
(for example, the instant velocity of a point moving in the manifold). As we learned
in the previous section, however, we must model everything in terms of coordinate
neighborhoods.
Let M be a smooth m-manifold and let x 2 M . Consider the set TQ Mx of all triples
.U; h; v/ where U is a neighborhood of x, h W U ! V be a diffeomorphism for
some V Rn open, and v 2 Rn .
Now introduce the following equivalence relation on TQ Mx : We put
.U; h; v/ .V; k; w/
h1 k
hŒW W kŒW ;
then
Dfh.x/ .v/ D w:
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2 Tangent vectors, vector fields and differential forms 293
(Recall that D denotes the total differential, see 3.2 of Chapter 3). It is easy to
verify that this is indeed an equivalence relation. The set of equivalence classes
of TQ Mx is denoted by TM x and its elements are called tangent vectors to M at
x. A representative of a
-equivalence class will be called a representative of a
tangent vector. The tangent vector represented by a triple .U; h; v/ will be sometimes
denoted by Œ.U; h; v/. When this gets too cumbersome, we will also refer to v as
the vector Œ.U; h; v/ in the coordinate system h W U ! V .
Note that by the lemma it immediately follows that TM x has a natural structure
of a R-vector space, and that moreover, this vector space is n-dimensional. In effect,
let U be an open neighborhood of x and let h W U ! V be a diffeomorphism onto
an open subset of Rn . Let
@ @
. ;:::; /: (*)
@h1 @hn
The reason for this notation is that if f W U ! R is a smooth function, in the spirit
of the chain rule, it makes sense to write
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294 12 Smooth Manifolds, Differential Forms and Stokes’ Theorem
where on the right-hand side, xi denotes the standard i ’th coordinate of Rn , as used
in Chapter 3.
It is also useful to notice that when U Rn is an open subset, x 2 U , we have a
canonical identification
Rn Š TU x
via
Dfx W TM x ! TN f .x/
This definition is correct by the chain rule (in Euclidean spaces) and Dfx is linear by
linearity of differentials (in Euclidean spaces). Additionally, note that it generalizes
the definition of total differential 3.2 of Chapter 3 when we identify the tangent
space of an open subset of Rn at every point with Rn .
If we have a real C 1 -function f W U ! R from some U M open, we usually
write df .x/ instead of Dfx . From this point of view, df can also be viewed as a
C 1 - 1-form (see 2.3 below). Similar statements, of course, hold for C r and smooth
functions. In particular, it is useful to note that if h W U ! V is a coordinate system
at x 2 M , and h D .h1 ; : : : ; hn /, then
.dh1 ; : : : ; dhn /
Dfx W TN f .x/ ! TM x ;
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2 Tangent vectors, vector fields and differential forms 295
y 7! Dhy .v.y// 2 Rn
resp.
is a C r map where the right-hand side uses the identification of the tangent spaces
of an open subset of Rn at the end of Section 2.1. C 1 vector fields and k-forms are
also called smooth.
It is useful to note that if h W U ! V is a smooth coordinate system at some
point x 2 M , h D .h1 ; : : : ; hn /, then immediately from the definition, the vector
space of all smooth vector fields on U is
X
n
@
f fi j fi W U ! R smooth functionsg;
i D1
@hi
X
n
f fi dhi j fi W U ! R smooth functionsg:
i D1
Thus, the smooth vector field or 1-form is completely determined by the n-tuple of
smooth functions .f1 ; : : : ; fn /, and vice versa, the functions fi are determined by
the vector field (resp. differential form) and the coordinate system.
Using Proposition 3.4 of Chapter 11, we can extend this to k-forms. The space
of all smooth k-forms on U is isomorphic to
X
f fi1 ;:::;ik dhi1 ^ ^ dhik j fi1 ;:::;ik W U ! R smoothg;
1i1 <<ik n
and the smooth functions fi1 ;:::;ik are completely determined by a smooth k-form.
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296 12 Smooth Manifolds, Differential Forms and Stokes’ Theorem
resp.
then this uniquely determines a smooth vector field resp. smooth k-form on M . In
other words, smooth vector fields and k-forms can be described by a collection of
local descriptions in the charts of an atlas.
Analogous statements are, of course, true with “smooth” replaced by C r .
This correspondence, of course, sends the identity to the identity, and .f ıg/ .!/ D
g .f .!//. Thus, we conclude that differential forms are contravariant in smooth
maps (in the sense of 1.2 of Chapter 11). There are, of course, analogous statements
for smooth replaced by C r .
It may be surprising that vector fields are neither covariant nor contravariant
in smooth maps: One can see this by realizing that vectors are covariant, while
smooth functions are contravariant. Vector fields can be made, however, covariant
in diffeomorphisms: Let f W M ! N be a diffeomorphism and let v be a smooth
vector field on M . Then we can define a smooth vector field f w on N by
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2 Tangent vectors, vector fields and differential forms 297
2.4.1 Comment
The meaning of the symbols f and f here is related to, but not quite the same as
in 1 of Chapter 11. Note that, for example, in the current situation, f is not a linear
map. Nevertheless, using the same symbol in both situations is quite standard in
this case.
The attentive reader has noticed a similarity of this material with our remarks on
substitution in differential equations. In fact, much of what we observed in Section 7
of Chapter 6 can be done coordinate-free. Let us make this concrete in one aspect,
which will be instructive as a contrast with what we will do with differential forms:
Proof. Let k W U1 ! V1 be any coordinate system at x and consider the vector field
k v. We can treat this vector field as a system of differential equations on V1 : For a
smooth function f W .a; b/ ! V1 , the equation is
X n
@f
f 0 .t/ D .k v/.f .t//i : (*)
i D1
@xi
Now we know that this system has a smooth solution in a neighborhood of a point
of V1 . Specifically, consider vectors w2 ; : : : ; wn 2 Rn such that
X
n
.0; a2 ; : : : ; an / D k.x/ C ai wi
i D2
f .t/ D .t; a2 ; : : : ; an /
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298 12 Smooth Manifolds, Differential Forms and Stokes’ Theorem
satisfies the equation (*). Additionally, by our assumption (**), the map is regular
at 0, so by the Inverse Function Theorem 7.3 of Chapter 3, there exists an open
neighborhood V of 0 such that the restriction jV W V ! ŒV is a diffeomorphism.
Now put
U D k 1 ŒV
and
h D . 1 k/jU : t
u
d W
k .M / !
kC1 .M /: (1)
X X
n
@fi 1 ;:::;ik
D dhj ^ dhi1 ^ ^ dhik :
1i1 <<ik n j D1
@hj
Lemma. The formula (2) does not depend on the choice of coordinate system.
d.fg/ D f dg C gdf:
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3 The exterior derivative and integration of differential forms 299
X n
@hi
dhi D dgj :
j D1
@gj
Now differentiating
X
@hi1 @hik
dfi1 ;:::;ik ::: dgj1 ^ ^ dgjk (4)
@gj1 @gjk
X
@hi1 @hik
fi1 ;:::;ik ::: dgj1 ^ ^ dgjk : (5)
@gj1 @gjk
and then multiply by dgj1 ^ ^ dgjk . However, by the Leibniz rule, we may
differentiate fi1 ;:::;ik and the partial derivative factors separately, and the key point is
that when we differentiate
@hip
;
@gjp
@2 hip
:
@gjp @gjp0
In the resulting sum, however, each such term will appear twice, with the attached
dgjp and dgjp0 terms swapped. Thus, by the rules of computation in the exterior
algebra, the two terms in each such pair appear with opposite signs, and hence
cancel out. t
u
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300 12 Smooth Manifolds, Differential Forms and Stokes’ Theorem
Lemma. We have
d ı d D 0 W
k .M / !
kC2 .M /:
@2 fi1 ;:::;ik
dh` ^ dhj ^ dhi1 ^ ^ dhik ;
@hj @h`
but each of these terms appears twice with j and ` in opposite orders, and therefore
with opposite signs, and hence the entire expression vanishes (of course, the terms
with j D ` vanish immediately). t
u
d d d
0 .M / !
1 .M / ! !
n .M / (*)
d ı d D 0:
The sequence (*) is called the de Rham complex of the smooth manifold M , and is
denoted by
.M /. A k-form ! is called closed if
d! D 0
! D d:
(We consider the 0-form 0 exact.) Then the set of all closed k-forms is a vector
subspace of
k .M / which is denoted by Z k .M /, and the set of all exact k-forms is
then a vector subspace of Z k .M / which is denoted by B k .M /.
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4 Integration of differential forms and Stokes’ Theorem 301
bk .M / D dim.HDR
k
.M //
and call this the k’th Betti number of M (it can, of course, be infinite, see
Exercise (17)).
Betti numbers are fundamental characteristics of manifolds. For example, they
are computable in practice, they turn out to be topological invariants, which
means that two homeomorphic manifolds have the same Betti numbers. Also, Betti
numbers can be defined for topological manifolds, and in fact, for all topological
spaces. This leads to an area of mathematics called algebraic topology (see, for
example, [3, 13, 14, 20]). Unfortunately, in this text, a systematic treatment of Betti
numbers would take us too far afield, and we will confine ourselves to a few basic
exercises (Exercises (11), (12) (13), (14), (15), (16), (17)).
Proof. To prove the first statement (existence), take a smooth atlas .Ui ; hi / such
that a form !i as required exists for the restriction of our orientation to Ui (such an
atlas exists by the definition of orientation). Now take a smooth partition of unity ui
subordinate to the cover Ui , and put
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302 12 Smooth Manifolds, Differential Forms and Stokes’ Theorem
X
!D ui !i :
i
To prove the second statement, let !, determine the same orientation. Choose a
smooth atlas .Ui ; hi /. Then !jUi D fi dh1 ^ ^ dhn , jUi D gi dh1 ^ ^ dhn .
Define k.x/ D fi .x/=gi .x/ when x 2 Ui . t
u
4.2 Integration
! D f dx1 ^ ^ dxn
(recall 2.4).
Lemma. The number (2) does not depend on the choice of the atlas .Ui ; hi / (subject
to the given conditions).
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4 Integration of differential forms and Stokes’ Theorem 303
Now let .Ui ; hi /i 2I , .Ui0 ; h0i /i 2I 0 be two atlases as in the statement of the lemma.
First, note that by the (finite) additivity of the integral, we may assume I D I 0 ,
Ui D Ui0 . We may still have hi ¤ h0i , but the invariance of the integral under this
choice follows from (3). t
u
Remark: Note that our notation is slightly inconsistent. In (2), we should display
the orientation of the manifold M . In (1), on the other hand, we assume the standard
orientation of Rn , i.e. the orientation defined by the n-form dx1 ^ ^dxn . A reversal
of orientation results, of course, in a reversal of sign.
4.3.1
Let M be an oriented smooth n-manifold. By a region with corners in M we
mean a compact subset K M such that for every x 2 K X K ı , there exists
an orientation-preserving coordinate system h W U ! V at x in M such that
V D .1; 1/n
or
(We use the symbol S n for the n-th Cartesian power of a set S here to reduce
the chance of confusion.) A special case worth pointing out is the case when one
always has k 1. In this case, we call K a compact n-dimensional submanifold
with boundary. Note that then our coordinate system gives K X K ı the structure of
a .k 1/-dimensional compact submanifold of M .
cj W Rn1 ! Rn
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304 12 Smooth Manifolds, Differential Forms and Stokes’ Theorem
Then define
Z
@K
XX
k Z
D .1/j ui .h1
i cj / (*)
i 2F1 j D1 h0;1/.k1/ .1;1/.nk/
XX
k Z
C .1/j ui .h1
i cj / :
i 2F2 j D1 .1;0i.k1/ .1;1/.nk/
It can be proved that the expression (*) does not depend on the choice of atlas with
the properties required above. However, this is a bit tedious and we will omit the
proof, as it is not needed for proving Stokes’ Theorem. When stating the theorem in
the next paragraph, we will simply assume that an atlas as above has been chosen.
It is worth noting, however, that in the special case of a compact n-dimensional
submanifold with boundary, it follows that the integral defined by (*) coincides with
Z
@K
.h1
i / .dx2 ^ ^ dxn / 2 ƒ
n1
.T .@K/x / :
(The minus sign comes from the fact that the added first vector of the ordered
basis representing the orientation of TMx should point “outside” from the boundary,
which, in our setup, happens to be in the negative direction.)
n1 .M /. Let K be a region with corners in M and let .Ui ; hi / and ui be chosen
as in 4.3. Then
Z Z
D d: (*)
@K K
Proof. The statement and the proof are both straightforward generalizations of
our treatment of Green’s Theorem. (In fact, the part of the proof dealing with
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4 Integration of differential forms and Stokes’ Theorem 305
Y
n
KD haj ; bj i;
j D1
Then
@f
d D .1/j C1 dx1 ^ ^ dxn :
@xj
(Note that on the right-hand side, the summands corresponding to coordinates other
than the j ’th coordinate vanish.)
Now in the general case, one proves the theorem by considering each of the
summands 4.3.2 (*) separately, applying the case of the cube to the smooth
.n 1/-form
ui .h1
i cj / :
h0; 1ik
h1; 1i.nk/:
h1; 0i.`1/
h0; 1i
h1; 1i.n`/
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306 12 Smooth Manifolds, Differential Forms and Stokes’ Theorem
d W
0 .U / !
1 .U /
is then identified with a map grad from the space of smooth functions on U to the
space of Rn -valued smooth functions (or, equivalently, n-tuples of smooth func-
tions) on U . The corresponding case of the Stokes Theorem is the “Fundamental
Theorem of Line Integrals” which says that for an oriented piecewise smooth curve
L represented by W ha; bi ! Rn , we have
Z
.II/ grad.f / D f ..b// f ..b//: (*)
L
(Note that our current setup is slightly different, to get a special case of Theorem 4.4,
we would have to formulate (*) on smooth 1-manifolds rather than piecewise
smooth curves, but both statements are equally easy to prove - see Exercise (20).)
Smooth .n 1/-forms can also be identified with smooth 1-forms and smooth
n-forms can be identified with smooth functions Pusing the Hodge -operator. For a
function F W U ! Rn , denote by the 1-form Fi dxi . Then we put
div.F / D .d .//;
(In this form, this integral is also known as flux.) Then the Stokes Theorem takes the
form
Z Z
F D div.F /:
@K K
.d/:
In coordinates, we obtain
@F3 @F2 @F1 @F3 @F2 @F1
curl.F / D ; ; :
@x2 @x3 @x3 @x1 @x1 @x2
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5 Exercises 307
Observe that the right-hand side may be interpreted as a sum of line integrals of the
second kind.
5 Exercises
(1) Prove that the definitions of a manifold and a smooth manifold would remain
equivalent if we require the coordinate maps hx to be homeomorphisms.
(2) Prove in detail that the definition given in Example 1.4 (4) really specifies a
smooth manifold and that the inclusion S n RnC1 is a C 1 -map.
(3) Prove that the function used in the proof of Theorem 1.5 is smooth.
(4) Recall the example of the manifold S n from the last section. For x 2 S n ,
construct an isomorphism of vector spaces
x W T .S n /x Š fw 2 Rn jx w D 0g
such that for every smooth map f W RnC1 ! RnC1 which satisfies f ŒS n S n
we have a commutative diagram
x
T .S n /x Rn
Df jS n Df
f .x/
T .S n /f .x/ Rn :
(5) Recall the notion of Lie bracket of smooth vector fields from 7.5 of Chapter 6.
Let us generalize this notion to vector fields on manifolds. In other words,
let u, v be vector fields which on some open set U with smooth coordinates
h1 ; : : : ; hn are given by
X
n
@ X @
n
uD fi ; vD gi
i D1
@hi i D1
@hi
X n
@gj @fj @
Œu; v D fi gi :
i;j D1
@hi @hi @hj
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308 12 Smooth Manifolds, Differential Forms and Stokes’ Theorem
v 7! v.e/:
(8) Prove that if G is a Lie group, then the vector space g of left-invariant smooth
vector fields on G forms a sub-algebra of the Lie algebra of all smooth
vector fields discussed in Exercise (5) in the sense that the Lie bracket of two
left invariant vector fields is left invariant. This g is called the Lie algebra
associated with the Lie group G, and can be shown to encode a large part of
the Lie group structure of G. (For further reading, see for example [9, 10].)
(9) Find two smooth 1-forms !, on R2 such that for every x 2 R2 we have
!.x/; .x/ ¤ 0 and there does not exist any non-empty open set U R2 and
W U ! R2 with D !jU . Compare with 2.5. [Hint: use the exterior
derivative.]
(10) Prove that for a smooth k-form ! and a smooth `-form ,
(11) Generalize the proof of Lemma 3.1 to prove that for a smooth map
f W M ! N and a smooth k-form ! 2
k .N /, we have
d.f !/ D f .d!/:
f W Z k .N / ! Z k .M /
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5 Exercises 309
which restricts to
f W B k .N / ! B k .M /
f W HDR
k
.N / ! HDR
k
.M /:
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310 12 Smooth Manifolds, Differential Forms and Stokes’ Theorem
M D f.x; z/ 2 C
S 1 jx 2 =z 2 Rg:
Prove that M is not orientable. [Hint: Consider the immersion and submersion
f W R
R ! M given by .x; t/ 7! .xe it ; e 2 it /. Prove that a 2-form hdxdy 2
2 .R2 / D f ! for a 2-form ! 2
2 .M / must satisfy h.0; 1/ D h.0; 0/ and
that therefore ! cannot be nowhere vanishing.]
(19) Consider, on S n1 , the smooth n 1-form
X
n
!D .1/i C1 dx1 ^ ^ dxi 1 ^ dxi C1 ^ ^ dxn :
i D1
Prove that
Z
! ¤ 0:
s n1
Conclude that bn1 .S n1 / 1. [Hint: use Stokes’ Theorem, the Hodge *
operator and spherical coordinates.]
(20) Prove the Fundamental Theorem of Line Integrals, 4.5 (*). [Hint: After
composing with the map , it becomes essentially a special case of the
Fundamental Theorem of Calculus for the Riemann integral, but a little bit
of care is needed since L is only piecewise smooth.]
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There are some extremely important concepts in complex analysis which we did
not cover in Chapter 10, and which ultimately lead up to several other areas of
mathematics. First of all, quite a bit more can be said about conformal maps. Under
very general conditions, one open subset of C can be mapped holomorphically
bijectively onto another. We prove one such result, the famous Riemann Mapping
Theorem. In many situations, such maps can even be written down explicitly. Those
are the Schwartz-Christoffel formulas, which have applications in cartography, as
the basic condition on mappings in cartography is to be conformal (since distortion
of distances in a topographical map is generally considered more allowable than
distortion of angles). Yet, the Schwarz-Christoffel formulas also lead to elliptic
integrals, which are “inverse” to elliptic functions (see for example [11]).
A major topic not covered in Chapter 10 is the question of “multi-valued holo-
morphic maps” such as, for example, the natural logarithm on C X f0g (or, for that
matter, elliptic integrals). What is the appropriate theoretical underpinning for such
functions? It turns out that now is the right moment for us to study such questions,
since we have already learned about manifolds. In this chapter, we will study com-
plex manifolds of complex dimension 1, which are called Riemann surfaces. It turns
out that the right way of thinking about multivalued functions on an open subset U
of C is as functions defined on a certain Riemann surface which is a covering of U
(not to be confused with open covers as studied in 1.1 of Chapter 9). In the process of
developing this concept, we will also learn a lot more about complex integration (we
will develop, for example, integration of holomorphic functions along continuous
paths and will show that if two paths are homotopic, i.e. one can be continuously
deformed to another, the integrals are the same). At the same time, we will also
explore striking ways in which complex differential forms behave on Riemann
surfaces, which will greatly enhance our understanding of complex integration.
Finally, we will see that methods of complex analysis extend even to functions
which are not holomorphic, generalizing, for example, the Cauchy formula to func-
tions which are continuously differentiable but not holomorphic. These methods will
be very useful in Chapter 15 below, where we will construct compatible complex
structures on oriented surfaces with Riemann metrics.
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312 13 Complex Analysis II: Further Topics
As is the case with the concept of manifolds, the study of coverings has a close
connection with algebraic topology, which we will not explore here in detail. We
will, however, briefly introduce the concept of the fundamental group and give two
examples in Exercises (15) and (16). For more information on Riemann surfaces,
we recommend the book [6], and for a very concise yet informative study of the
fundamental group and coverings in an abstract topological setting, [13]. For very
interesting ventures to higher dimensions, [8] may be an excellent source.
Proof. Let us study the singularity of the function f .1=z/ at z D 0. If this singularity
is removable, then f is bounded, and hence constant by Liouville’s Theorem 5.1 of
Chapter 10, contradicting our assumptions. If f .1=z/ has a pole of order k > 1 at 0,
then for " > 0 sufficiently small, there exists, by Theorem 6.3.3 of Chapter 10, a ı >
0 such that 1=f .1=z/ a has exactly k zeros in
.0; "/ for every a 2
.0; ı/. Note
that these k zeros may include zeros of order > 0, but not for ı sufficiently small,
since otherwise the holomorphic function .1=f .1=z//0 would have zeros arbitrarily
close to 0, and hence would be constantly 0 by Theorem 4.4 of Chapter 10. However,
if the k zeros are all different, this contradicts injectivity of f . Finally, if f .1=z/
has an essential singularity at 0, let f .0/ D A. Then by the Holomorphic Open
Mapping Theorem 6.3.4 of Chapter 10, for every r > 0 there exists an " > 0
such that f j
.0; r/ takes on every value in
.A; "/. On the other hand, applying
Proposition 6.2 of Chapter 10 to f .1=z/, we see that there are (infinitely many) z
with jzj > r such that f .z/ 2
.A; "/ which, again, contradicts injectivity.
We have concluded that f .1=z/ has a pole of order 1 at z D 0. Then the
function .f .z/ A/=z is holomorphic and bounded on C, and hence is constant
by Liouville’s Theorem 5.1. t
u
az C b
fA .z/ D
cz C d
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1 The Riemann Mapping Theorem 313
and we assume
det.A/ ¤ 0:
fA ı fB D fAB ;
and thus all Möbius transformations are bijective maps C [ f1g ! C [ f1g.
We will understand that better in Section 3 below. While in the formalism we
introduced, Möbius transformations with c ¤ 0 are, by definition, not holomorphic
functions on C, they can be useful in mapping injectively holomorphically certain
open subsets of C onto one another (see Exercises 1, 2).
Proof. First note that by Theorem 6.3.3 of Chapter 10, f 0 .z/ ¤ 0 for all z 2
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314 13 Complex Analysis II: Further Topics
Schwartz’s Lemma again, f .z/ D cz with jcj D 1, but the assumption that f 0 .0/ is
a positive real number then implies c D 1. t
u
for all z 2 U .
0 .z/
Proof. Since U is simply connected, the function has a primitive function on
.z/
U , which we will denote by Ln..z//. This function is determined up to an additive
constant. But using the chain rule and the product rule, we find that
0
e Ln..z//
D 0;
.z/
so this function is constant. The additive constant can therefore be chosen in such a
way that
Proof. First of all, note that uniqueness follows from Corollary 1.1.3, since if f1 , f2
were two maps satisfying the conclusion of the Theorem, then .f1 /1 f2 would be a
holomorphic automorphism of
.0; 1/ with positive real derivative at 0.
To prove existence, we will first prove that there exists an injective holomorphic
map f W U !
.0; 1/ with f .z0 / D 0 where f 0 .z0 / is a positive real number. In
effect, let a … U . Apply Lemma 1.2 to the function .z/ D z a. Thus, we have a
function Ln.z a/ such that
e Ln.za/ D z a:
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1 The Riemann Mapping Theorem 315
Now let
h.z/ D e Ln.za/=2 :
Then
h.z/2 D z a on U ;
h.z/ ¤ ˙h.t/:
By the Holomorphic Open Mapping Theorem 6.3.4 of Chapter 10, there is an r > 0
such that
.h.z0 /; r/ hŒU :
Therefore,
hŒU \
.h.z0 /; r/ D ;:
jh.z/ C h.z0 /j r;
and in particular,
2jh.z0 /j r:
First, note that the denominator is non-zero. Clearly, we have f0 .z0 / D 0. From the
chain rule, in fact,
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316 13 Complex Analysis II: Further Topics
which shows that jf0 .z/j 1 for z 2 U . Of course, strict inequality must hold by
the Holomorphic Open Mapping Theorem 6.3.4 of Chapter 10.
Now let N be the supremum of all the values f 0 .z0 / over the set S of all injective
functions f W U !
.0; 1/ which satisfy f .z0 / D 0 and f 0 .z0 / > 0. (Note that
a priori, one may have N D C1.) There exists, however, a sequence .fn /n of
functions in S such that
Clearly, the sequence fn is uniformly bounded and by Theorem 3.7 of Chapter 10,
is also equicontinuous on every compact subset of U . By Theorem 6.3 of Chapter 9
(a consequence of the Arzelà-Ascoli Theorem), there exists a subsequence .fin /n
which converges uniformly on every compact subset K U . Denote the limit
function by f . We know by Weierstrass’s Theorem 3.6 of Chapter 10 that f is
holomorphic, f 0 .z0 / D N (and thus N < 1), and jf .z/j 1 for every z 2 U , but,
again, a strict inequality must arise by Theorem 6.3.4 of Chapter 10. We will now
show that f is injective. In effect, let z1 2 U . Then the functions fin .z/ fin .z1 /
have no zero in U Xfz1 g, and hence f .z/f .z1 / has no zero in U Xfz1 g by Hurwitz’s
Theorem 6.3.6 of Chapter 10. Since z0 was arbitrary, f is injective as claimed.
We claim that the function f W U !
.0; 1/ is onto. Assume, for contradiction,
that w0 2
.0; 1/ X f ŒU . Let
f .z/ w0
.z/ D :
1 w0 f .z/
Let, again,
so that
.z/ D .g.z//2 :
Let
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2 Schwartz-Christoffel formula 317
1.3 Comments
1. It is clear why the case U D C must be excluded in the statement of the Riemann
Mapping Theorem: By Liouville’s Theorem 5.1 of Chapter 10, any bounded
holomorphic map defined on C is constant.
2. Excluding the case of U D C, as already remarked, Proposition 2.5 of Chapter 10
provides a converse to the Riemann Mapping Theorem when stated for real-
regular images of convex open sets. Perhaps much more importantly, however,
Proposition 2.5 of Chapter 10 serves as a source of examples for the Theorem.
While our definition of a simply connected set above precisely fits the proof of
the Theorem, it is not a condition which is easy to verify. On the other hand,
constructing real injective regular maps on convex sets, as in Proposition 2.5 of
Chapter 10, is easy (for example, see Exercise (4)).
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318 13 Complex Analysis II: Further Topics
.z z0 /˛ D e ˛Ln.zz0 / (2.1.2)
can then be defined on any open half-plane P whose boundary contains z0 , and
inspection shows that this function can be extended to a bijective continuous
function mapping P onto a closed angle of value ˛.
Let f .w/ be a holomorphic function defined in an open neighborhood U of a
point w0 2 C, let f .w0 / D z0 and let f 0 .w0 / ¤ 0. Define
g.w/ D .f .w/ z0 /˛ :
g.w/.w w0 /˛
1
!˛
X
g.w/ D anC1 .w w0 / n
;
nD0
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2 Schwartz-Christoffel formula 319
of w, and additionally
ŒP \ R D ŒLk1 [ L0 :
Let this image be the interval hs; ti where s < 0 < t. Now applying the argument
of the previous paragraph to the holomorphic isomorphism ˆ from the set
Int.ŒP [ fz j z 2 ŒP g/
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320 13 Complex Analysis II: Further Topics
Now the statement about the holomorphic extension of the function (2.3.1)
follows from Lemma 2.2 (applied to this same function g). t
u
Z wY
k
g.w/ D C .u wi /ˇi du C D (2.4.1)
0 i D1
Y
k
h.w/ D g 0 .w/ .w wi /ˇi (*)
i D1
Ci Ci 1 D ˇi :
Arg.wi / C Arg.w/
Arg.w wi / D ˙ C : (**)
2 2
Therefore, for w on the circle segment between wi and wi C1 ,
!
Y
k X
k
Arg .w wi /ˇi D Qi C . ˇj /Arg.w/=2 D Qi C Arg.w/
i D1 j D1
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3 Riemann surfaces, coverings and complex differential forms 321
for some constant Qi . When passing wi in the clock-wise direction, one of the C
signs in (**) changes into a , which shows
Qi Qi 1 D ˇi :
We see then that the argument of the function h.w/ of (*) is constant on the unit
circle. Thus, the holomorphic function h.w/ on U maps the unit circle into a set of
the form
S D fte ib j t > 0g
for b constant (a ray). Applying the Maximum Principle 6.3.5 of Chapter 10 to the
holomorphic functions
ib ib
e h.w/e ; e h.w/e ;
Comment: The numbers wi are not determined by Theorem 2.4 or any of the
above discussion. They are difficult to determine analytically except in a few very
special situations (see Exercises (6), (7)).
We will now use what we learned about complex analysis to discuss a partial
“complex analog” of some of the material of Chapter 12. While this may seem like
an abstract exercise, it actually turns out to be an extremely useful device, which
will enhance greatly our understanding of topics already covered, such as Möbius
transformations, simply connected open subsets of C, and even primitive functions.
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322 13 Complex Analysis II: Further Topics
h1
x f hf .x/
hx Œ.f 1 ŒUf .x/ / \ Ux f 1 ŒUf .x/ \ Ux Uf .x/ C
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3 Riemann surfaces, coverings and complex differential forms 323
Now it is pretty much obvious from the definition that the Möbius transformations
of Subsection 1.1 are holomorphic automorphisms of CP 1 , and it is not difficult to
check that they are the only ones (see Exercise (10)).
Moreover, for an open set U C, note that a meromorphic function on U is
precisely the same thing as a holomorphic map U ! CP 1 . Because of this, one
extends this to call a meromorphic function on a Riemann manifold † a holomorphic
map f W † ! CP 1 .
Here is another example: Let a; b be complex numbers linearly independent over
R. Introduce an equivalence relation
on C where x1 C iy1
x2 C iy2 is x1
x2 D ka, y1 y2 D `b where k; ` are integers. The set E of equivalence classes
with respect to this equivalence relation is called an elliptic curve. (The use of the
term “curve” here stems from algebraic geometry, where one develops methods for
defining geometric objects, called varieties, over general fields. A 1-dimensional
variety is called a curve. A non-singular curve over the field C is then, in particular,
a Riemann surface.)
Denote the equivalence class of z 2 C by Œz, an element of an equivalence class
is called its representative. Clearly, we have a projection
WC!E
given by
.z/ D Œz:
We may define a metric E by letting the distance of two classes Œz0 , Œt0 be
min jz tj
where z 2 Œz0 , t 2 Œt0 . The reason the minimum exists is that the subset
L D fka C `b j k; ` 2 Zg
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324 13 Complex Analysis II: Further Topics
is discrete. The projection is then continuous. There exists, therefore, an " > 0
such that
.0; "/ \ L D f0g. Then for any z 2 C, j
.z; "/ is a homeomorphism
onto
.Œz; "/. Thus, the inverses of these restrictions can be taken for an atlas,
making E a Riemann surface.
Meromorphic functions on E are the same data as doubly periodic functions on
C. Such functions are called elliptic functions. See Exercise (8) for one method by
which examples of elliptic functions can be constructed.
3.3 Coverings
3.3.1
Let † be a Riemann surface. A holomorphic map W T ! †, where T is another
Riemann surface, is called a covering if for every z 2 †, there exists an open
neighborhood Vz such that 1 ŒVz is a disjoint union of open subsets Ui , i 2 I ,
such that for each i , the restriction
jUi W Ui ! Vz
W Uf ! U
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3 Riemann surfaces, coverings and complex differential forms 325
Proof. Let At be an open interval containing the point t 2 h0; 1i such that !ŒAt \
h0; 1i is contained in a fundamental neighborhood. By Theorem 5.5 of Chapter 2,
h0; 1i is covered by finitely many of the open intervals At . Denoting their end points
by 0 D t0 < t1 < < tk D 1, each of the images !Œhti ; ti C1 i is contained in a
fundamental neighborhood. We can prove by induction on i that a lift !Q i of !jh0; ti i
with end point x exists and is unique: in fact, assuming this for a given i , !Q i exists,
let V be a fundamental neighborhood containing !Œhti ; ti C1 i, and let Vj be the open
subset given by the definition of a covering which is mapped homeomorphically to
V by the restriction i of the projection, and has the property that !Q i .ti / 2 Vj . Then
for t 2 hti ; ti C1 i, define
Clearly, this extends !Q i to the required !Q i C1 , and further this extension is uniquely
determined, since i is a homeomorphism. Now we can put !Q D !Q k , and we have
both existence and uniqueness.
Regarding the homotopy, let h W ! ' . We shall construct a lift of this
homotopy to T . Note that we already know the lift exists and is uniquely determined
by applying the path lifting theorem separately to the path h.‹; a/ with each
fixed a. However, we must prove that this lift hQ W h0; 1i
h0; 1i ! T is
continuous. To this end, we must repeat, to some extent, our above argument
for paths: The set h0; 1i
h0; 1i is compact, and hence is covered by finitely
many rectangles hs; s 0 i
ht; t 0 i the closures of whose images lie in fundamental
neighborhoods. Taking the finite sets of all such s; s 0 and t; t 0 , we obtain partitions
0 D s0 < s1 < < s` D 1, 0 D t0 < t1 < < tm D 1 where the h-image of
each rectangle hsi ; si C1 i
htj ; tj C1 i is in a fundamental neighborhood Ui;j . For each
j , we then prove by induction on i that hjh0; Q si i
htj ; jj C1 i is continuous; indeed,
suppose the statement is true for a given i (and a fixed j ). Then by the connectedness
of intervals and the induction hypothesis, hŒfs Q i g
htj ; tj C1 i is contained in one of
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326 13 Complex Analysis II: Further Topics
the disjoint open sets which, by , map homeomorphically onto Ui;j . Inverting the
homeomorphism, we obtain the statement for i C 1.
Q 1/ is constant in s, note that 1 Œf!.1/g is discrete, and a
To see that h.s;
continuous function from a connected space to a discrete space is constant. t
u
Remark: It is useful to note that the proof of this theorem was purely topological
and did not make any use of the holomorphic structure.
ƒk .TMx / ˝R C
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3 Riemann surfaces, coverings and complex differential forms 327
dz D dx C i dy: (**)
z D x C iy:
The right-hand side of (**) is then a differential 1-form with complex coefficients,
so we may integrate it over piecwise continuously differentiable curves L in C.
When integrating the left-hand side of (**) over L, we mean, on the other hand, the
corresponding complex line integral. This is, then, the same thing as treating dz as
a complex-valued 1-form. Using complex multiplication, we then have additional
complex 1-forms ! D f .z/dz for a complex continuously real-differentiable
function f .z/. A line integral of the complex-valued 1-form ! is then the same
thing as the complex line integral as treated earlier, thus explaining in this way a
complex line integral as an integral of a complex-valued 1-form.
dz D z0 .t/dt
where z0 denotes the complex derivative (note that we only need to make sense of
this on an open subset of C). This means that 1-forms on a coordinate system, which
can be given as
f .z/dz;
dF D !: (*)
To see that this is the right generalization, note that on an open set U C, indeed,
dF D f .z/dz is equivalent to F 0 .z/ D f .z/, see Exercise (12). Note that therefore
by what we proved in Chapter 10, it immediately follows that a primitive function
to a holomorphic 1-form (if one exists) is necessarily holomorphic.
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328 13 Complex Analysis II: Further Topics
Even if a primitive function does not exist, note that the construction 3.3.2
immediately generalizes to give, for any holomorphic 1-form ! on a Riemann
surface †, a covering
W †! ! †:
Again, †! D †
C as a set, and the topology has basis consisting of sets 3.3.2 (*),
where V † is open, and F is a primitive function of ! on V .
dz D dx i dy
dz D z0 .t/dt :
are preserved by holomorphic change of coordinates. Such forms are called 1-forms
of type .1; 0/, resp. of type .0; 1/. In fact, note that if we define, for ! D f .z/dz C
g.z/dz with f; g smooth on U ,
! D f .z/dz C g.z/dz;
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3 Riemann surfaces, coverings and complex differential forms 329
Note that in this notation, the Cauchy-Riemann equations for a function f can be
expressed simply by
@f
D 0: (3.5.1)
@z
Regarding the exterior differential, one has, of course, for a complex continuously
(real)-differentiable function f ,
@f @f
df D dz C dz:
@z @z
@f
@.f !0 / D dz ^ !0 ;
@z
@f
@.f !0 / D dz ^ !0 ;
@z
d D @ C @:
One readily verifies that @ and @ are invariant under a change of holomorphic
coordinate (see Exercise (13)). Because of that, @ and @ are well-defined on any
Riemann surface †.
Note that on a compact Riemann surface, there may exist non-trivial holomorphic
1-forms. For example, the form dz obviously determines a well-defined holomorphic
1-form on any elliptic curve as defined in Subsection 3.2. Compare this with
Exercise 9 which asserts that every holomorphic function on a compact Riemann
surface is constant. In fact, note that if † is a compact Riemann surface, then the
space
1Hol .†/ embeds canonically into the de Rham cohomology with complex
coefficients
1
HDR .†; C/ D HDR
1
.†/ ˝R C:
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330 13 Complex Analysis II: Further Topics
1
HDR .†; C/ Š
1Hol .†/
1Hol .†/:
Let us remark that the 1-form dz, of course, pulls back to any open subset U C,
and hence also to any covering W V ! U . We shall simplify notation by denoting
dz D d.z ı / also by dz, thus defining “complex integration” of functions on
any covering † equipped with a covering W V ! U where U C is an open
subset. Since every point z 2 V has an open neighborhood which is mapped by
holomorphically bijectively onto an open subset of U , a complex derivative of
holomorphic functions f W V ! C is then also defined, as is the concept of a
primitive function of f on open subsets of V .
Z Z
!D !:
L M
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3 Riemann surfaces, coverings and complex differential forms 331
In effect, find again 0 D t0 < t1 < < tk D 1 such that LŒhti ; ti C1 i, M Œhti ; ti C1 i
for each chosen i are contained in a fundamental neighborhood of the covering, and
use the properties of primitive functions.
But then since L; M are homotopic, .z1 ; K1 / D .z2 ; K2 / by Theorem 3.3.4,
which proves our statement. t
u
Note that it would be quite difficult to prove this directly using the techniques of
Chapter 10, in particular since there is no theory of line integrals of the second kind
over continuous paths: we have really used the force of Theorem 3.3.4 here.
However, for open subsets of C, we can go even further. Recall the definition of
a simply connected open set from Subsection 1.2.
3.6.3 Theorem. For a connected open set U ¨ C, the following are equivalent:
(1) U is simply connected (i.e. every holomorphic function on U has a primitive
function).
(2) U is holomorphically isomorphic to
.0; 1/
(3) Let a; b 2 U . Then any two paths !; with beginning point a and end point b
are homotopic.
Proof. (1) implies (2) by the Riemann Mapping Theorem 1.2. (2) implies (3)
because
.0; 1/ is a convex set: We may define the homotopy simply by h.s; t/ D
t!.s/ C .1 t/.s/. To see that (3) implies (1), suppose that U is a connected
open subset of C satisfying (3). Let f be a holomorphic function on U . Let †
be the covering 3.3.2 corresponding to the primitive function of f , and let †0
be a connected component of U . By definition, the restriction of the projection
0 W †0 ! U is a covering. We claim, in fact, that it is a holomorphic isomorphism.
By Theorem 3.3.4, and the fact that U is path-connected, 0 is onto. Thus, if it is
not a holomorphic isomorphism, it cannot be injective, i.e. there must be two points
x; y 2 †0 with 0 .x/ D 0 .y/. But †0 is connected, and since it is a manifold, also
path-connected, so there is a path ! in beginning point x and end point y. Then the
projection 0 ı ! in U has the same beginning point and end point 0 .x/ D 0 .y/,
but cannot be homotopic to the constant path by Theorem 3.3.4, since its lift ! has
a different beginning point and end point.
The contradiction proves that 0 is a holomorphic isomorphism; the second
coordinate of 01 .z/ is then a primitive function of f on U . t
u
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332 13 Complex Analysis II: Further Topics
.0; 1/, C or CP 1 , but we shall not prove this here (see, however, Exercise (17)).
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4 The universal covering and multi-valued functions 333
4.1.1 Lemma. If .Œ!/ D .Œ/ (i.e. ! and have the same end point x) and
Œ! ¤ Œ (i.e. ! and are not homotopic), then for any convex open subset V
hx ŒUx ,
UŒ!;V \ UŒ;V D ;:
! ..hx /1 ı L/ ..hx /1 ı L/ ' ..hx /1 ı L/ ..hx /1 ı L/;
!'
D ! ..hx /1 ı L/
where L is a linearly parametrized line segment with beginning point hx .x/ and
end point hx .y/. To prove that UŒ;W UŒ!;V , let M be a linearly parametrized
line segment in W with beginning point hy .y/ and end point hy .z/. We need to
prove that
Œ .h1
y ı M / 2 UŒ!;V : (*)
.h1 1 1
y ı M / ' ! .hx ı L/ .hy ı M /:
Now we have
.h1 1 1 1
x ı L/ .hy ı M / D hx ı .L .hx ı hy ı M //:
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334 13 Complex Analysis II: Further Topics
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4 The universal covering and multi-valued functions 335
Proof. Let x 2 †. Q Let ! be a path in † Q with beginning point xe0 and end point x.
By Theorem 3.3.4, there is a unique lift of the path ı ! to T with beginning
point y0 . Let .x/ be the end point of . (Note in fact that this definition is forced by
the path lifting property, which already implies uniqueness.) On the other hand, also
note that our definition of .x/ did not depend on the choice of the path !, since
any two such paths are homotopic as † Q is simply connected. Because of this, if
U is a connected fundamental open neighborhood of a point z 2 † for both the
coverings and , and if Ui (resp. Uj ) is the open disjoint summand of 1 ŒU
(resp. 1 ŒU ) such that 0 D jUi ! U (resp. 0 D jUj ) and which contains
x (resp. .x/), then jUi is given by the formula 01 ı 0 , which shows that
is a covering with such fundamental neighborhoods Uj . (Note: if y 2 T is not
in the connected component of the base point, then it won’t be in the image of ,
so the fundamental neighborhood of y can be chosen to be the whole connected
component.) t
u
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336 13 Complex Analysis II: Further Topics
Note that then, in particular, the multivalued function based at a point x0 does have
a well-defined “value” at the point x0 .
Multivalued holomorphic functions based at x0 form an algebra in the sense
that they contain (ordinary) holomorphic functions (a holomorphic function f is
identified with the multi-valued function f ı ), and have well-defined operations
of addition and multiplication. Much more is true, of course, for example if f is a
multi-valued holomorphic function based at x0 and g W C ! C is an ordinary holo-
morphic function, then there is a well-defined multivalued holomorphic function
g ı f based at x0 .
Note that by Corollary 4.2.3, the choice of † Q does not matter in the sense
that multi-valued holomorphic functions defined via any other based holomorphic
Q by a preferred bijection, namely
universal covering are related to those defined via †
the one induced by the based holomorphic isomorphism between † Q and T , and that
this bijection preserves all the operations in sight. It is important to note, however,
that unless † is simply connected, there is no preferred way of identifying the
algebras of multivalued holomorphic functions based at different base-points of †.
Examples of multi-valued holomorphic functions on Riemann surfaces can be
obtained from holomorphic 1-forms !: Note that we have a primitive function F
of ! well-defined on any connected component of the covering †! , and hence,
by Theorem 4.2.1, on the universal cover. This is referred to as the multi-valued
primitive function of !. Note that a discussion of base points is not so important here,
since no matter how we choose base points, two multi-valued primitive functions of
the same holomorphic 1-form will differ by a constant. In particular, for connected
open sets U C, we have a well-defined notion (up to additive constant) of a multi-
valued primitive function based at z0 2 U of a given multi-valued function based
at z0 .
For example, the multi-valued primitive function of
1
f .z/ D
z z0
on C X fz0 g with value equal to 0 at the base point which is chosen to project to
z0 C 1 2 C X fz0 g is called the multivalued logarithm ln.z z0 /. Choosing an
arbitrary ˛ 2 C, we then obtain the multivalued function
.z z0 /˛ D e ˛ ln.zz0 /
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4 The universal covering and multi-valued functions 337
4.2.5 Example
The behavior of multi-valued functions can be quite complicated. Consider the
multivalued function
on U D C X f0; 1g. Assume, for simplicity, a; b > 0 to be real numbers. (Note that
there exists unique multi-valued functions za , .z 1/b based at any chosen point
0 < z0 < 1 whose values at the base point are positive real numbers.)
Now let F be the multi-valued primitive function on U (Let, for example, the
value of F at the base point zQ0 , .Qz0 / D z0 , be 0.) Now let K be the circle with
center 0 and radius z0 (and beginning point z0 ) oriented counter-clockwise, and let
L be the circle with center 1 and radius 1 z0 (and beginning point z0 ) oriented
counter-clockwise.
Let ! be a concatenation of m copies of K and n copies of L (in any fixed order),
and ze1 be the end-point of the lift !Q to the universal covering with beginning point
zQ0 . Then one immediately sees that
Let us now examine the behavior of the function F : First note that the integrals
Z z0 Z 1
AD za .z 1/b dz; B D za .z 1/b dz
0 z0
actually exist in the sense of ordinary real analysis, and are equal to (finite) positive
real numbers. Additionally, the integrals of (1) over a circle with radius " and center
0 or 1 goes to 0 with " ! 0. Because of this, if K e is a lift of K to the universal
covering with beginning point zQ1 as above, we have, denoting the end point by zQ2 ,
while if e
L is a lift of L to the universal cover with beginning point zQ1 as above and
end point zQ3 , we have
Note that the operations (3), (4) do not commute: if we begin at zQ1 and follow first
K and then L, the value of the primitive function increases by
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338 13 Complex Analysis II: Further Topics
while following L first and then K beginning from the same point zQ1 gives an
increase of
These two values are in general not equal. Because of this, it is not true, contrary to
what one may naively expect, that F .z/=f .z/ would be a single-valued function on
U (in the sense that it would be a composition of an ordinary holomorphic function
on U with ). Note that we also see that the end points of the lifts of K L and
L K to the universal covering with the same beginning point are, in fact, different.
Up to normalization, the function F belongs to a family of functions
called hypergeometric functions; they are, in some sense, the “simplest” multi-
valued holomorphic functions on a connected open subset of C for which this
phenomenon occurs.
then Œ! is the inverse of Œ! with respect to . Thus, the set 1 .†; x0 / with the
operation is a group in the sense of Appendix B, 3.1. This group is called
the fundamental group of † with base point x0 . There are many interesting and
deep connections between the fundamental group and coverings, which we cannot
explore in this text, in part because we do not develop the theory of groups in any
substantial way. After filling in the necessary algebra, say, in [2], the reader can find
more information in [6, 13, 20].
There is, however, one connection between the fundamental group and the
universal cover which is too beautiful and striking to pass up. Consider a based
universal cover
Q xQ 0 / ! .†; x0 /
.†;
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4 The universal covering and multi-valued functions 339
f
Σ̃ Σ̃
π π
ˆ W ! 1 Œfx0 g
ˆ.f / D f .xQ 0 /:
‰ W 1 .†; x0 / ! 1 Œfx0 g
as follows: Let ! be a path in † with beginning point and end point x0 . Let !Q
be a path in † Q which is the unique lifting of ! with beginning point xQ 0 (see
Theorem 3.3.4). Then let ‰.Œ!/ be the end point of !. Q By Theorem 3.3.4, this
does not depend on the choice of the representative ! of the class Œ! 2 1 .†; x0 /.
The following result can often be used to compute the fundamental group (see
Exercise (15)).
Theorem. The maps ˆ and ‰ are bijections. Moreover, the composition ˆ1 ı ‰
is a homomorphism (hence isomorphism) of groups.
Proof. The fact that ˆ is bijective is a special case of the universality Theorem 4.2.1.
To show that ‰ is onto, recall that † Q is connected, and hence path-connected. Let
y 2 1 Œfx0 g and let be a path in † Q from xQ 0 to y. Put ! D ı . Then ‰.Œ!/ D
y. To prove injectivity, note that the just mentioned is unique up to homotopy
since † Q is simply connected, and composing with gives uniqueness of Œ!.
To prove that ˆ1 ı ‰ is a homomorphism of groups, let , ! be paths in † with
beginning points and points x0 and let , Q with beginning point
Q !Q be their lifts to †
xQ 0 . Let, on the other hand, O be the lift of to † Q whose beginning point is the end
point xQ 1 of !.Q Now let f be a deck transformation which sends xQ 0 to xQ 1 . Then by
uniqueness of path lifting, f ı Q D . O In particular, if we denote the end point of Q
by xO 0 and the end point of O by xO 1 , then
f .xO 0 / D xO 1 :
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340 13 Complex Analysis II: Further Topics
We see that
so
while
4.4 Comment
The reader no doubt noticed that the concepts of covering, universal covering, and
fundamental group do not use the structure of a Riemann surface very substantially.
They can, indeed, be defined for more general topological spaces. In order for the
nice theorems we presented to be true, however, some “local assumptions” about
the topological spaces involved must be included. The book [20] contains an easily
accessible discussion of coverings in a more general topological context. One case
which works very well is the case of smooth (or even topological) manifolds.
Definition 3.3.1, Theorem 3.3.4, Theorem 4.1, Theorem 4.2.1, the definition of
fundamental group in 4.3 and Theorem 4.3 remain vaild if we replace “Riemann
surface” by “smooth manifold” (resp. “topological manifold”) and “holomorphic
isomorphism” by “diffeomorphism” (resp. “homeomorphism”).
Yet, the case of Riemann surfaces, which we discussed above, is particularly
striking, and in this context, coverings were first discovered by Riemann.
We are now ready to extend Cauchy’s formula (Theorem 3.3 of Chapter 10) to the
case of any continuously (real)-differentiable function. Let us write an integration
variable
D s C it
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5 Complex analysis beyond holomorphic functions 341
Z k Z
1 .@f =@/dsdt 1 X f ./d
C D f .z/: (5.1.1)
U z 2 i j D1 Lj z
Proof. It is actually almost the same as the proof of Theorem 3.3 of Chapter 10.
Using the language of Subsection 3.5, we may rewrite (5.1.1) as
Z k Z
1 f ./d 1 X f ./d
d C D f .z/: (*)
2 i U z 2 i j D1 Lj z
On the other hand, for " > 0 small, if we denote by K the boundary of
.z; "/
oriented counter-clockwise, then Green’s Theorem 5.4 of Chapter 8 gives
Z k Z Z
1 f ./d 1 X f ./d 1 f ./d
d C D : (**)
2 i U X
.z;"/ z 2 i j D1 Lj z 2 i K z
When " ! 0, the right-hand side tends to f .z/ by the same argument as in the proof
of Theorem 3.3 of Chapter 10. So it remains to prove that
Z
.@f =@/dsdt
lim D 0;
"!0
.z;"/ z
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342 13 Complex Analysis II: Further Topics
D D
.0; 1/:
(We could, of course, equivalently work on any other disk.) Where needed, we may
extend such functions to C by 0. Note also that for z 2 D, using polar coordinates,
we have
1
./ D 2 Lq .D/ for every q < 2:
z
P W Lp .D/ ! L1 .C/
defined by
Z
1 f ./dsdt
.P .f //.z/ D :
D z
We will also need another version of this operator, defined by the formula
Z
1 1 1
.P1 .f // D f ./ dsdt:
C z
is in Lq .CX
.0; 2jzj// for every q > 1, and thus P1 .f / is defined for any function
f 2 Lp .C/, p > 2, and produces a (not necessarily bounded) complex function
defined everywhere on C.
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5 Complex analysis beyond holomorphic functions 343
Z
@P1 .f .z// 1 f ./ f .z/ f ./ f .0/
D dsdt;
@z C . z/2 2
(5.2.4)
@P1 .f .z//
D f .z/ f .0/:
@z
Proof. Let us first prove the statement for P . Using polar coordinates, one easily
proves the identity
Z
1 dsdt
D z; for z 2 D. (1)
D z
f ./ f .z/
j zj˛
We must show that (3) converges to 0 with
z ! 0. The integral (3) is certainly
finite, and without loss of generality, z D 0. Now a substitution D
z shows that
(3) is proportional to j
zj˛ , and hence tends to 0 with
z ! 0, as needed.
Proving the continuity of
@P .f .z//
@z
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344 13 Complex Analysis II: Further Topics
Z
1 f ./ f .z/
z f ./ f .0/
z
dsdt
C z z
z
z
The Lebesgue dominated convergence argument can then be applied on the set
C X .
.z; 2
z/ [
.0; 2
z//;
lim f .z/ D 0
z!1
and that there exists a function A.z/ with continuous first (real) partial derivatives
and compact support such that
@f
D Af:
@z
Proof. Assume without loss of generality that the support of A.z/ is contained in D.
Put F .z/ D f .z/e .P .A//.z/ . Using Lemma 5.2.1, we compute
@F @f .z/
D e B.z/ f .z/A.z// D 0:
@z @z
Finally, we will prove two easy inequalities involving the operator P , which will
also be useful in Section 5 of Chapter 15:
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5 Complex analysis beyond holomorphic functions 345
8jjf jj1
j.P .f //.z/j :
1 C jzj
(2) For every p > 2 there exists a constant Cp such that if f 2 Lp .C/, then
Thus, for jzj 1, we may use r D 2 to show that the left-hand side of (*) is less than
or equal to 4 . For jzj > 1, the idea is to integrate 1=j zj over the intersection of
.z; 1 C jzj/ X
.z; jzj 1/ (**)
with the smallest angle with center z which contains D. As already remarked, the
integral of 1=j zj over (**) is 4 , so the integral over the intersection of (**) with
an angle of size ˛ will be
2˛:
2
2arcsin.1=jzj/ ;
jzj 1 C jzj
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346 13 Complex Analysis II: Further Topics
ˇZ ˇ Z 1=q
ˇ f ./ f ./ ˇ dxdy
ˇ ˇ
dxdy ˇ jzj jjf .z/jjp
ˇ
C z C j. z/j
q
Z 1=q
dudv
D jjf .z/jjp jzj.2=q/1:
C j. 1/jq
6 Exercises
(1) Prove that a Möbius transformation maps an open disk, an open half-plane
or the complement of a closed disk onto an open disk, open half-plane or the
complement of a closed disk. Prove furthermore that for any two subsets of C[
f1g of any two of the above three types, there exists a Möbius transformation
mapping one onto the other.
(2) Let w0 2
.0; 1/. Consider the Möbius transformation
z w0
f .z/ D :
1 w0 z
is simply connected.
(5) Find an elementary function which maps the set fz 2
.0; 1/ j Re.z/ >
0; Im.z/ > 0g bijectively holomorphically onto
.0; 1/. [Hint: Find, in this
order, holomorphic isomorphisms of the set described onto an open half-disk,
an open quadrant, an open half-plane,
.0; 1/.]
(6) Show that if the polygon P is a triangle, then in Theorem 2.4, the points
w1 ; w2 ; w3 can be chosen to be any three points on the unit circle which occur in
this order when the circle is oriented counter-clockwise. [Hint: Using the maps
of Exercise (2) and rotations, show that there is a holomorphic automorphism
of
.0; 1/ which extends holomorphically to an open set containing
.0; 1/,
and maps a given choice of points w1 ; w2 ; w3 to any other such given choice.]
(7) Determine a choice of the points wi when P is a regular k-gon.
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6 Exercises 347
(8) Using the Schwartz-Chrisfoffel formula, write down an explicit formula (with
one free parameter) for a function f mapping bijectively holomorphically the
upper half-plane on a rectangle. Such formulas are called elliptic integrals.
Using complex conjugation (similarly as in Lemma 2.3), prove that the inverse
function g extends to a meromorphic function on C, which is doubly periodic,
with periods equal to the sides of the rectangle (such functions are called
elliptic functions). For information on elliptic function, the reader may look
at [11].
(9) Prove that every holomorphic function on a compact Riemann manifold is
constant.
(10) Prove that the Möbius transformations are the only holomorphic automor-
phisms of CP 1 . [Hint: Use Proposition 1.1.1.]
(11) Prove that non-constant meromorphic functions on CP 1 are precisely rational
functions, i.e. functions of the form p.z/=q.z/ where p.z/, q.z/ are polyno-
mials, q.z/ not identically zero. [Hint: Multiply (resp. divide) such a function
f .z/ by the product of all factors .zzi /ki where zi is a pole (resp. zero)of order
ki in C, (infinitely many zeroes or poles would mean f is a constant 0 or 1 by
the Uniqueness Theorem 4.4 of Chapter 10). Then we may assume without loss
of generality that the restriction of f to C has neither zeroes nor poles. Now
if f .1/ ¤ 1, then f is bounded on C, while if f .1/ D 1, then 1=f .z/
is bounded on C. In either case, f is constant by Liouville’s Theorem 5.1 of
Chapter 10.]
(12) Prove in detail that for U C an open set, F .z/ is a primitive function for the
1-form f .z/dz with f holomorphic if and only if F .z/ is a primitive function
of f .z/.
(13) Prove in detail that the definitions of the operators @, @ on differential forms
on a Riemann surface is invariant under holomorphic change of coordinates.
(14) Prove that the function e z , considered as a holomorphic map C ! C X f0g, is
a covering and that, in fact, it is the universal covering of C X f0g.
(15) From Exercise (14), construct an isomorphism
1 .C X f0; g; x0 / ! Z
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348 13 Complex Analysis II: Further Topics
(19) Recall the concept of a Lie group from Chapter 12, Exercise (6). Prove that the
fundamental group of a Lie group is commutative (see Comment 4.4). [Hint:
the concatenation of paths is homotopic to the point-wise product, using the
group operation.]
(20) Prove that if W ! G is a covering and G is a Lie group (cf. Comment 4.4)
with both G and connected, then can be given a structure of a Lie group
such that is a homomorphism of groups.
(21) Define for f 2 Lp .D/, p > 2,
Z
1 f ./dsdt
.Q.f // D :
D z
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The aim of this chapter is to give a glimpse of the main principle of the calculus of
variations which, in its most basic problem, concerns minimizing certain types of
linear functions on the space of continuously differentiable curves in Rn with fixed
beginning point and end point. For further study in this subject, we recommend [7].
We derive the Euler-Lagrange equation which can be used to axiomatize a large
part of classical mechanics. We then consider in more detail the possibly most
fundamental example of the calculus of variations, namely the problem of finding
the shortest curve connecting two points in an open set in Rn with an arbitrary given
(smoothly varying) inner product on its tangent space. The Euler-Lagrange equation
in this case is known as the geodesic equation. The smoothly varying inner product
captures the idea of curved space. Thus, solving the geodesic equation here goes a
long way toward motivating the basic techniques of Riemannian geometry, which
we will develop in the next chapter.
1.1
y W ha; bi ! Rn (*)
as a function with the property that the function defined as the derivative of y on
.a; b/ and as the respective one-sided derivatives at a and b is everywhere defined
and continuous on ha; bi.
Now consider the vector space V D Va;b;p;q of all continuously differentiable
function (*) such that
y.a/ D p; an y.b/ D q
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350 14 Calculus of Variations and the Geodesic Equation
L D L.t; x1 ; : : : xn ; v1 ; : : : ; vn / W ha; bi
R2n ! R
S WV !R
given by
Z b
S.y/ D L.t; y.t/; y0 .t//dt:
a
Note that S is continuous when we consider the metric on V given by the norm
(Here we may choose any of the usual norms on Rn , for example the maximum
one.) However, in the kind of formal investigation we are going to do, even this will
play only a peripheral role.
for all continuously differentiable functions h such that h.a/ D h.b/ D 0. Then
f 0.
Proof. Suppose f is not identically zero. Then f .t0 / ¤ 0 for some t0 2 .a; b/.
Suppose, without loss of generality, f .t0 / > 0. Since f is continuous, there exists
an " > 0 such that f .t/ > 0 for all t 2 .t0 "; t0 C "/. Now let u be a continuously
differentiable function which is positive on some non-empty interval contained in
.t0 "; t0 C "/, and 0 elsewhere (we may use the “baby version” of smooth partition
of unity 5.1 of Chapter 8). Then
Z b
f .t/h.t/dt > 0:
a
t
u
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1 The basic problem of the calculus of variations, and the Euler-Lagrange equations 351
ˇ ˇ
@L ˇˇ d @L ˇˇ
D :
@xi ˇxDy;vDy0 dt @vi ˇxDy;vDy0
but there is some danger in such a notation, since in the partial derivatives, we must
treat yi , yi0 as formal symbols plugged in for the arguments xi , vi of L, while the
derivative by t is the actual total derivative by the independent variable t.
Proof of the theorem: Choose any continuously differentiable function h W
ha; bi ! R, such that h.a/ D h.b/ D 0. Consider the real function of n variables
Z b
ˆh .u1 ; : : : ; un / D L.t; y.t/ C uh.t/; y0 .t/ C uh0 .t//dt:
a
If the functional L has an extreme at y, then ˆh has an extreme at o, and since it has
continuous partial derivatives by the chain rule everywhere, we must have
@ˆh .o/
D 0:
@ui
Z
1 @L.t; y.t/ C uei h.t/; y 0 .t/ C uei h0 .t//
b
D uh.t/dt
u a @xi
Z !
b
@L.t; y.t/; y0 .t/ C uei h0 .t// 0
C uh .t/dt
a @vi
for some 0 < ; < 1. On the right hand side, we used the Mean Value Theorem 3.3
of Chapter 3 twice. Note that the u factor cancels out, and using h.a/ D h.b/ D 0
and integration by parts in the second integral, we get
Z b
@L.: : : / d @L.: : : /
D h.t/dt:
a @xi dt @vi
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352 14 Calculus of Variations and the Geodesic Equation
1.3 Comment
The main idea of the proof resembles the idea of the total differential of a function
of finitely many variables (see Exercise (1) below for a more concrete statement). It
may seem we got something for free: how come we can find extremes of functionals
on a space of continuously differentiable functions as easily as extremes of functions
of finitely many variables? There is, however, one major catch: with the space V not
being compact (not even locally), there is no guarantee an extreme of the functional
S on V exists at all! Therefore, Theorem 1.2 is not nearly as strong as it may seem,
giving only candidates for a possible extreme. Similarly as in the case of functions
of finitely many variables, we call these candidates critical functions. Highly non-
trivial methods are generally needed to show that a given critical function is in fact
an extreme (we will see an example of that below).
and hence
p
L.t; x; v/ D 1 C v2 :
Therefore, we have
@L v
Dp ;
@v 1 C v2
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2 A few special cases and examples 353
y0
p D K;
1 C y 02
or
.1 K 2 /y 02 D K 2 :
Then we have
d X @L.y.t/; y0 .t//
n
@L.y.t/; y0 .t// 00
L.y.t/; y0 .t// D yi0 C yi ;
dt i D1
@xi @vi
Thus, we obtain
!
d X n
@L 0
L yi D 0;
dt i D1
@vi
or in other words
X n
@L 0
y LDK (2.2.1)
i D1
@vi i
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354 14 Calculus of Variations and the Geodesic Equation
We may choose units such that the mass of the car is 1, as is the acceleration of
gravity. Then the potential energy at the point .0; s/ is s, and hence by conservation
of energy, at a point .t; x/, the kinetic energy is .s x/. Thus, if the component of
the velocity in the t direction is w, we have
1 2
w .1 C x 02 / D s x;
2
and hence
s
2.s x/
wD ;
1 C x 02
or
s
1 C v2
L.t; x; v/ D 1=w D :
2.s x/
which yields
p
1 D K 2.s x/ .1 C x 02 /
or
1 D 2K 2 .1 x/.1 C x 02 /:
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2 A few special cases and examples 355
In the basic setup of Newtonian mechanics, the particles have masses mi , and the
kinetic energy is
X
n
1
2 C v3i 1 C v3i /:
2 2 2
m.v3i (2.3.2)
i D1
2
P1 2
A kinetic energy formula of this form, i.e. essentially the form 2
mv , is referred
to as a standard kinetic term.
The potential energy term is more variable. Assuming the particles act on one
another by gravity, Newton’s law of gravity gives potential energy
X mi mj
Gv (2.3.3)
u 2
i <j uX
t .x
3i k x3j k /
2
kD0
X
n
i .x3i 2 ; x3i 1 ; x3i /: (2.3.4)
i D1
According to the recipe (2.3.1), the (original) Lagrangian is obtained by taking the
standard kinetic term (2.3.2), and subtracting the potential terms (2.3.3), (2.3.4),
thus getting
X
n
1
D 2
m.v3i 2 C v3i 1 C v3i /
2 2
i D1
2
X mi mj X
n
C Gv C i .x3i 2 ; x3i 1 ; x3i /:
u 2
i <j uX i D1
t .x x3j k /2
3i k
kD0
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356 14 Calculus of Variations and the Geodesic Equation
Lagrange’s principle states that the equation of motion is given by the critical
function for this Lagrangian on a time interval ha; bi with given positions at the
times a and b, i.e. that it is subject to the Euler-Lagrange equations 1.2. We will not
prove this here. In fact, a mathematical “proof” in this setting is not to be expected:
we are referring to a system of physical particles. What could be proved, however,
is that Lagrange’s equations are equivalent to Newton’s.
Observe that in the presence of the standard kinetic term (2.3.2), the Hamilto-
nian (2.2.1) of 2.3.1 has the physical meaning of the total energy of the system,
which, indeed, should be conserved by the law of conservation of energy.
The Lagrangian mechanics setup may seem like nothing new, since it only recov-
ers Newton’s equations, and, in fact, is even less general, since it requires a conser-
vative force field. However, the Lagrangian turns out to be extremely beneficial for
generalizations. In fact, most of modern physics uses the Lagrangian formalism.
Let us return to mathematics. Perhaps the single most important example of the
Euler-Lagrange equation is the geodesic equation in a Riemann metric (although it
should be pointed out that the equation does have a physical meaning, describing in
fact the motion of a light ray in a gravity field in Einstein’s general relativity).
hu; vig
W ha; bi ! U
by the formula
Z b q
sg ./ D h 0 .t/; 0 .t/ig dt: (3.1.1)
a
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3 The geodesic equation 357
Thus, using this convention, the components of the function will be written with
superscripts, i , i D 1; : : : ; n, and the formula (3.1.1) above will assume the form
Z b q Z b q
sg ./ D gij 0i 0j D gij ..t// 0i .t/ 0j .t/dt: (3.1.3)
a a
The convention (3.1.2) may seem unreasonably restrictive, but turns out adequate
in the types of formulas we will encounter. It is known as (one version of) the
Einstein convention. When two quantities share an index as a subscript in one
and a superscript in the other (and summation over all permissible values is to
be performed), we call the quantities coupled. We can see already in (3.1.3) in
comparison with (3.1.1) that the Einstein convention can make formulas more
explicit. In the next chapter, when talking about the more general context of
manifolds, we will talk about tensors, and will give the Einstein convention a deeper
interpretation.
We see immediately that the Euler-Lagrange equation for the functional (3.1.3)
will be a pain because of the square root in the Lagrangian. This problem has a
surprisingly simple solution, which, at first, cannot possibly seem right: simply omit
the square root! Thus, we will consider the functional
Z b
Sg ./ D gij 0i 0j : (3.2.1)
a
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358 14 Calculus of Variations and the Geodesic Equation
The last step may seem to do nothing, but we will see later that it is useful to have the
quantity coupled to .x j /0 .x k /0 symmetrical in j; k (it will help eliminate a certain,
somewhat counterintuitive, quantity known as torsion).
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3 The geodesic equation 359
@L.x; x0 / @gj k j 0 k 0
D .x / .x / ;
@x i @x i
and hence the Euler-Lagrange equation becomes (after cancelling 2),
i 00 1 @gij @gik @gj k
gij .x / C C j .x j /0 .x k /0 D 0; (3.3.2)
2 @x k @x @x i
g ij gj k D ıki
where
ıki D 1 when i D k
D 0 otherwise
ji k D g i ` `j k ;
The symbols ijk or ji k are known as Christoffel symbols of the first resp. second
kind.
Parametrized curves satisfying the geodesic equation are called geodesics
parametrized by arc length, or simply geodesics. Let us keep in mind, however,
that geodesics are merely critical for the functional (3.2.1) of 3.2. We have not
proved that geodesics minimize the length of continuously differentiable curves
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360 14 Calculus of Variations and the Geodesic Equation
with given boundary points. In fact, this is false in general (see Exercise (7) (c)
of Chapter 15 below). Yet, for the sake of geometry, we are clearly interested at
least in some minimum length statement regarding geodesics, and it is important to
note that the variational tools we supplied do not give that. We will prove such a
statement in the next section using different methods.
The purpose of this section is to study geodesics in more detail, and eventually to
prove that locally they really are the curves of minimal length connecting two points
with respect to a given Riemann metric.
@y.t; x/
D f.t; y.t; x// (*)
@t
By 1.2 of Chapter 6, an analogue of Lemma 4.1.1 also holds for systems of higher
order differential equations. Applying this specifically to the case of the geodesic
equation (3.3.5) of 2.3, we obtain the following
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4 The geometry of geodesics 361
(Here on the left hand side, h‹; ‹i denotes the dot product, see Appendix A,
Section 4.3.) Then there exists a convex open neighborhood V of o 2 Rn , and a
unique smooth map W V ! U such that
(1) .o/ D P ,
(2) for each v 2 Rn , .vt/ considered as a function of t in an open neighborhood
of o in which vt 2 V is a g-geodesic parametrized by arc length (in the sense
of 3.3),
(3) @v .o/ D .v/.
The smooth map of Corollary 4.1.2 is often denoted by exp and called the
exponential map.
Let us first verify that solutions of the equation (3.3.5) of 3.3 are indeed parametrized
by arc length with respect to the Riemann metric g. While we argued in 3.2 that this
must be true for parametric curves minimizing the functional (3.2.1), note that we
have so far only proved that the solutions of (3.3.5) are critical. Hence, that argument
cannot be used rigorously.
.gij .x i /0 .x j /0 /0 D 0
Proof. Let us compute the Hamiltonian (2.2.1) of 3.2 for the Lagrangian (3.3.1)
of 2.3:
@L.x; x0 / i 0
.x / L.x; x 0 / D 2gij .x i /0 .x j /0 gij .x i /0 .x j /0 D gij .x i /0 .x j /0 :
@v i
Thus, the quantity whose constancy in t we are trying to prove is in effect the
Hamiltonian. Hence, our statement follows from 2.2. u
t
Note that the proof of Lemma 4.2.1 suggests multiplying the Lagrangian (3.3.1)
of 3.3 by a factor of 1=2, and calling it energy.
4.2.2
Now we will prove that when we shift a geodesic to a nearby geodesic, the
angle of the shift is also conserved, provided that we do not change the scale of
parametrization. More precisely, let solutions of the geodesic equation (3.3.5) of 3.3
depend on some smooth parameter u in the space of initial conditions, as in the proof
of Lemma 4.1.1. Let us assume further that
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362 14 Calculus of Variations and the Geodesic Equation
@gij .x i /0 .x j /0
D 0: (1)
@u
Note that by Lemma 4.2.1, it suffices to verify this condition at one point, and
the condition indeed means that we are not changing the scale of arc length
parametrization with u. Now let
@x
zD ;
@u
as, again, in the proof of Lemma 4.2.1.
Lemma. We have
Proof. Compute
@gij @x k @x i @x j @2 x i @x j @x i @2 x j
.gij .zi /.x j /0 /0 D C gij C gij : (3)
@x k @t @u @t @u@t @t @u .@t/2
@gij @x k @x i @x j @2 x i @x j
C 2gij D 0: (4)
@x k @u @t @t @u@t @t
Subtracting 1=2 times (4) from the right hand side of (3), we get
@gij @x k @x i @x j 1 @gij @x k @x i @x j @x i @2 x j
D k
k
C gij : (5)
@x @t @u @t 2 @x @u @t @t @u .@t/2
2 j
Using the geodesic equation (3.3.5) of 3.3 for @ x 2 , we see that the second term is
.@t/
equal to
This is equal to
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4 The geometry of geodesics 363
@gij @x k @x i @x j 1 @gij @x k @x i @x j
k
C
@x @t @u @t 2 @x k @u @t @t
by renaming variables, which shows that (5) is 0. t
u
Remark: In comparison with Lemma 4.2.1, we may ask if Lemma 4.2.2 has
a similarly conceptual proof (our proof was by calculation from the definition of
the Christoffel symbols). Such a conceptual proof indeed exists, and is related to
our comments in Sections 7 and 8 of Chapter 6: the condition (1) indicates that
the Lagrangian has an infinitesimal symmetry. By a similar but somewhat more
elaborate argument to the discussion in Chapter 6, this always implies a conserved
quantity known as a Noether current, which is the cause of the conservation law
proved in Lemma 4.2.2. Discussing this more systematically, however, exceeds the
scope of this text.
Let us now consider an open subset U Rn with a smooth Riemann metric g and a
point P 2 U . Choose an isometry as in Corollary 4.1.2, and let W V ! U ,
.0/ D P , be the corresponding exponential map. By the Inverse Function
Theorem 7.3 of Chapter 3, we may further assume that is a diffeomorphism
onto its image.
Sr D fx 2 Rn j jjxjj D rg:
Caution: It is not claimed, and, as we will see in the next section, certainly not
true in general, that would be an isometry!
Proof. We will use Lemma 4.2.2. Let xQ D x=jjxjj D x=r. Consider the geodesic
.t xQ /. By the definition of , and the fact that it is a diffeomorphism onto its image
when restricted to V , the space T ..Sr //.x/ is spanned by the vectors z.r/ of 4.2.2
with respect to the boundary condition change
x.0; u/ D P; (*)
0 xQ C uw
x .0; u/ D
jjQx C uwjj
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364 14 Calculus of Variations and the Geodesic Equation
where hx; wi D 0. The condition (1) of 4.2.2 is then satisfied (at t D 0 and hence,
by Lemma 4.2.1, for all t 2 .r; r/) by the fact that is an isometry. By (*),
sg .y/ r;
h W .V / ! R
given by
h.x/ D jj 1 xjj:
@h.x/
. /i (a)
@x i
We have
@h.x/ @h.x/
g ij D 1: (c)
@x i @x j
(Change coordinates so that one coordinate vector will be the derivative of (b) at
t D jj 1 .x/jj and the other, g-orthogonal coordinate vectors will be tangent vectors
at x to .Sjj 1 .x/jj /. Then the contributions to (c) in the new coordinates from all
but i D j D 1 will be 0, and the contribution from the first coordinate is 1 by the
fact that the geodesic (b) is parametrized by arc length, and is an isometry.) Hence,
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5 Exercises 365
where equality arises if and only if y0 .t/ is a positive multiple of a tangent vector
of a geodesic of the form (b) for y.t/ D x almost everywhere in t (and hence
everywhere, by continuity). t
u
5 Exercises
where
M W Va;b;o;o ! R
satisfies
lim M.h/ D 0
h!0
and
@L.t; y.t/; y0 .t// d @L.t; y.t/; y0 .t//
.Dy .t/i / D :
@xi dt @vi
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366 14 Calculus of Variations and the Geodesic Equation
(5) By reversing the coordinates in Example 2.2 (i.e. making the vertical coor-
dinate the independent and the horizontal coordinate the dependent variable),
find an alternate solution to the brachistochrone problem using the method of
Example 2.1.
(6) Find the critical functions for the functional
Z 1
S.u; v/ D ..u0 /2 C .v 0 /2 C u0 v 0 /dx:
0
(7) Prove in detail the parametric form (2.2.2) of the solution of the brachys-
tochrone problem.
(8) Prove that the formula (3.1.1) of 3.1 does not depend on the parametrization
of a piecewise continuously differentiable curve L.
(9) The hyperbolic plane is the upper half-plane of complex numbers, i.e. the set
H D fx C iy 2 C j y > 0g
C D fx C iy j x; y 2 Rg;
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The attentive reader probably noticed that the concept of a Riemann metric on an
open subset of Rn which we introduced in the last chapter, and the related material
on geodesics, beg for a generalization to manifolds. Although this is not quite as
straightforward as one might imagine, the work we have done in the last chapter
gets us well underway. A serious problem we must address, of course, is how the
concepts we introduced behave under change of coordinates. It turns out that what
we have said on covariance and contravariance in manifolds is not quite enough: we
need to discuss the notation of tensor calculus.
Additionally, it turns out that discussing geodesics in a Riemann metric directly
would cause us to copy many expressions over and over unnecessarily. There is a
natural intermediate notion which axiomatizes the Christoffel symbols of the second
kind directly, without referring to a Riemann metric. This gives rise to the concept
of an affine connection. In the presence of an affine connection, we can discuss
geodesics, but also the important geometric concepts of torsion and curvature. We
will show that vanishing of torsion and curvature characterizes, in an appropriate
sense, the canonical affine connection on Rn (the flat connection).
We will define the notion of a Riemann manifold, and show how it canonically
specifies an affine connection, known as the Levi-Civita connection. This will lead
us to the concept of curvature of a Riemann manifold. We will show that locally, a
Riemann manifold with zero curvature is isometric to an open subset of Rn . We will
also show that every oriented Riemann manifold in dimension 2 has a compatible
structure of a Riemann surface.
Although we make no reference to physics, the present chapter gives a good
rigorous foundation for the mathematics of general relativity theory. In fact, the
notation we use (writing out the indices in tensors) is closer to physics than is
customary in most mathematical texts. As we shall see, this notation does not
sacrifice rigor, and can make calculations with tensors more transparent by showing
explicitly which coordinates we are contracting.
To comment on the title of this chapter, by tensor calculus, one usually means
the basic development of tensor fields, their transformation under changes of
coordinates, and the covariant derivative. Riemannian geometry develops the same
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368 15 Tensor Calculus and Riemannian Geometry
1 Tensor calculus
h ˝ ˝ h Ty ; y 2 U (1.1.2)
„ ƒ‚ …
m C n times
@
v D vi
@hi
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1 Tensor calculus 369
(using the Einstein convention). From the point of view of tensor calculus, we write
v simply as v i . Note that composing the coordinate system h with where W U !
V is a diffeomorphism of open subsets of Rn , we have
@ @hj @
D
@. ı h/i @ i @hj
by the chain rule, and thus with respect to the coordinates . ı h/i , the coordinates
of v will be
D 1 .v 1 ; : : : ; v n /T :
Even though we have not specified coordinates, it is often customary to give a tensor
of type .m; n/ m different superscripts and n different subscripts, e.g.
Tji11ji22:::i m
:::jn :
The superscripts and subscripts are formal symbols each one of which refers simply
to a particular factor of (1.1.1). For example a tensor of type .2; 2/ may be then
denoted by
ij
Tk` :
This notation has immediate benefits. For example, the Einstein convention now
makes sense for tensors: for tensors T , S , by the symbol
we mean the image of S ˝ T under the map which applies the evaluation map
.TM x / ˝ .TM x / ! R
to the coordinates of S and T labeled by i . We stipulate that each index will occur
at most twice, but there may be multiple pairs of coinciding indices, in which case
we apply multiple evaluation maps: For example,
ij
Tk` Sijk` 2 R
makes sense for two tensors of type .2; 2/ at the same point x 2 M . This operation
is often referred to as contraction.
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370 15 Tensor Calculus and Riemannian Geometry
The other benefit is that we can easily talk about symmetric and antisymmetric
tensors: Recall that for two vector spaces V; W there is a canonical interchange map
V ˝ W ! W ˝ V; v ˝ w 7! w ˝ v:
A tensor
1
TM x ;TM x .IdTM x /:
ıji :
1.3 Comment
The reader probably noticed the difference between the way subscripts and
superscripts are used in the context of tensors on a Riemann manifold, and the
way we used them in the last chapter: in the last chapter, an index i stood simply
for the i ’th coordinate, where i is a number, and the Einstein convention was used
to sum terms where the same i occurs twice. In the context of tensors, no number
is plugged in for i , it simply is a label denoting which factor of the tensor product
we are working with, and the Einstein convention means an application of the
evaluation map.
Conveniently, these two points of view are somewhat interchangable: if we pick
@
a local coordinate system h, then we have a basis i of TM x , and a dual basis dhi
@h
of .TM x / , and the evaluation map can be indeed computed by summing products
of terms coupling a basis element with the corresponding element of the dual basis.
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2 Affine connections 371
Nevertheless, one must be careful to note that the coordinate-free tensor context
is more restrictive: The tensor notation should be used only for quantities which
are intrinsically coordinate-free. For example, let us take the Christoffel symbols
ijk . On an open set in Rn , the tangent space is canonically identified with Rn , so
we could certainly view ijk as a tensor of type .1; 2/. The trouble is, however, that
if we change coordinates, i.e. apply a diffeomorphism to another open subset of
Rn , this will not preserve the tangent space identification, and we find that it would
not preserve the tensor ijk we just defined, i.e. that for each choice of coordinates,
we would get a different tensor. Usually, this is expressed by saying that ijk is not
a tensor and transforms according to different rules (see Exercise (1) below). It
is more accurate, however, to say that there is no canonical tensor given by the
Christoffel symbols.
2 Affine connections
W.M /
W.M / ! W.M /;
.u; v/ 7! ru .v/
and
ru .f v/ D @u f v C f ru .v/: (2.1.2)
2.2 Locality
Perhaps the first thing to notice about affine connections is that they are “local” in
the following sense: the value of ru .v/ at a point x 2 M clearly depends only on
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372 15 Tensor Calculus and Riemannian Geometry
the value u.x/ and the value of v on the image of any continuously differentiable
oriented curve
W .a; a/ ! M
v D f i ei ; w D g i ei ;
@u f i .x/ D @u g i .x/:
2.3 Examples
1. The most basic example is the canonical connection in Rn : since the tangent
space of Rn is canonically identified with Rn , vector fields are canonically
identified with Rn -valued functions, and we may simply define the value of ru .v/
at x as the u.x/-directional derivative of v (considered as an Rn -valued function)
at x.
2. Let us now generalize this example in the spirit of the previous section. Let U be
an open subset of Rn and let gij be a Riemann metric on U . Define
@v j
ru .v/ D u i
ej C v j ijk ek (2.3.1)
@x i
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2 Affine connections 373
rx 0 .t / x 0 D 0; (2.3.2)
2.4.1
Let W ha; bi ! M be a continuously differentiable parametrized curve in a smooth
manifold M with affine connection r (as usual, we assume that 0 .t/ ¤ 0 for any
t 2 ha; bi and take the one-sided derivatives at the boundary points). Consider the
equation
r 0 .t / y..t// D 0 (*)
where y is a smooth vector field defined on Œha; bi. Clearly, we can treat this
problem locally, and hence we may work in a coordinate neighborhood U of M ,
where we have a smooth coordinate system h W U ! RN . Let ei D @ i . Writing
@h
y D x i ei ;
the equation (*) becomes a system of first-order linear differential equations in the
coefficients x i . Thus, by Theorem 1.3 of Chapter 7, there is a unique solution to the
equation (*) with given value
v D y..a// 2 TM .a/ :
This solution is called the parallel transport of the vector v along the parametrized
curve with respect to the affine connection r. It is important to note, however, that
performing parallel transport on a vector v D .a/ 2 TM .a/ along a parametrized
closed curve may produce a different vector v ¤ .b/ 2 TM .b/ D TM .a/ . This
is related to two quantities known as torsion and curvature associated with the affine
connection r, which we will discuss in the next section.
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374 15 Tensor Calculus and Riemannian Geometry
2.4.2 Geodesics
We can now also see that the concept of a geodesic generalizes to any smooth
manifold with an affine connection. In effect, if, in a local coordinate system h,
we write ei D @ i , and define Christoffel symbols of the connection r by
@h
then in this generalized sense, any affine connection in local coordinates is given
by the formula (2.3.1) of 2.3 (by the axioms of 2.1). We then see that the
“geodesic equation” (2.3.2) written in coordinates becomes a (non-linear) second-
order ordinary differential equation, and hence locally has solutions uniquely
determined by the value and derivative at a single point (by Corollary 4.1.2 of
Chapter 14).
Recall the vector space W.M / of smooth vector fields on M . We will prove the
following
ˆ W W.M /
W.M / ! W.M /
„ ƒ‚ …
k times
Proof. By the same reasoning as in 2.2, the value ˆ.u1 ; : : : ; uk /x depends only on
values of ui in an open neighborhood U of x. We may assume U to be a coordinate
neighborhood with a coordinate function h W U ! RN , and let ei D @ i . Then we
@h
may write
ui D i y j ej
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3 Tensors associated with an affine connection: torsion and curvature 375
1y
j1
.x/ k y jk .x/ˆ.ej1 ; : : : ; ejk /x
3.2
and
where Œu; v is the Lie bracket of the smooth vector fields u; v (see Section 7 of
Chapter 6, and Exercise (5).
Lemma. The functions T , R satisfy the hypotheses of Lemma 3.1, and hence define
smooth tensor fields Tijk , Rijk
`
. Furthermore, both of these tensors are antisymmetric
in the coordinates i; j .
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376 15 Tensor Calculus and Riemannian Geometry
3.3 Example
The connection defined in Example 2.3 2 has zero torsion. This immediately follows
from the fact that
Compare this to the beginning of Subsection 3.3 of Chapter 14, where we specifi-
cally defined the Christoffel symbols in such a way so as to make (3.3.1) true.
In fact, more generally, we see from the comments made in 2.4.2 and formula
(2.3.1) of 2.3 that any affine connection has zero torsion if and only if, in local
coordinates, it satisfies (3.3.1) in the sense of 2.4.2.
Proof. Clearly, the Euclidean connection has torsion and curvature 0, and hence the
existence of the coordinate system h W U ! Rn with the specified properties implies
that r is torsion and curvature free on U .
On the other hand, consider a connection r on M which is torsion and curvature
free on an open neighborhood of x. Choose a basis e1 ; : : : ; en of TM x . Let W
.a1 ; a1 / ! M , .0/ D x, 0 .0/ D e1 be a geodesic with respect to r. Now
denote the parallel transport of e2 along also by e2 at each point t1 2 .a1 ; a1 /. Let
t1 W .a2 ; a2 / ! M be a geodesic with t1 .0/ D .t1 /, t01 .0/ D e2 . Note that we
may assume the number a2 > 0 is independent of t1 because of smooth dependence
on geodesics on boundary conditions (the argument of 2.4 extends verbatim to this
situation). By the same argument, we may also consider as a smooth function
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3 Tensors associated with an affine connection: torsion and curvature 377
W .a1 ; a1 /
.a2 ; a2 / ! M:
@ @
Œ ; D0 (1)
@t1 @t2
@
ei D ; (2)
@ti
i D 1; 2. By the fact that r has 0 curvature, parallel transports along the curves t1 ;‹
and ‹;t2 with constant t1 resp. t2 therefore commute. We conclude in particular that
re2 .e1 / D 0;
re1 .e2 / D 0;
Hence, in fact,
re1 .e1 / D 0;
W .a1 ; a1 /
.ak ; ak / ! M
such that if we define (2), then (3) is true for all i; j 2 f1; : : : ; kg. If k < n,
denote the parallel transport of ekC1 to any of the points .t1 ; : : : ; tk / by the
curves .t1 ; : : : ti 1 ; ‹; ti C1 ; : : : tn / (with only one ti non-constant) by ekC1 . Smooth
dependence on boundary conditions implies that is a smooth function of the k C 1
variables t1 ; : : : ; tkC1 on some set
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378 15 Tensor Calculus and Riemannian Geometry
.a1 ; a1 /
.akC1 ; akC1 /
and applying the above argument to individual pairs of coordinates gives (3) for
i; j 2 f1; : : : ; k C 1g.
Thus, we may assume k D n. But then is locally the inverse of a local
coordinate system on M at x (by the Inverse Function Theorem), and (3) implies
that this coordinate system carries the connection r to the Euclidean connection
2.31, as claimed. t
u
4 Riemann manifolds
g.u; v/ D gij ui v j
is positive-definite (and hence defines a real inner product). A smooth manifold with
a Riemann metric is called a Riemann manifold. The fact that we considered an inner
product on TM x (as opposed to TMx ) is merely a convention: we claim that given
a Riemann metric gij , there exists a unique tensor of type .2; 0/ denoted by gij such
that
gij g j k D ıik ;
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4 Riemann manifolds 379
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380 15 Tensor Calculus and Riemannian Geometry
Therefore,
g.ru v; w/
D 12 .@u .g.v; w// C @v .g.u; w// @w .g.u; v// g.Œu; w; v/ g.Œv; w; u// :
Hence, ru v is determined by g.
Now we will prove existence. We will first treat the case when M D U is
an open subset of Rn . In this case, consider the connection (2.3.1) constructed in
Example 2.32. We already know from Example 3.33 that this connection is torsion
free. To verify that this connection is compatible with the metric g, by the chain
rule, it suffices to verify the condition (2) in the case when u D ei , v D ej , w D ek .
Thus, we need to show that
which translates to
@gij
D kij C j i k ;
@x k
which follows directly from equation (3.3.3) of Chapter 14.
Now let M be an arbitrary smooth Riemann manifold, and let .Ui / be a
coordinate cover of M . Then by what we just proved, and by locality of connections,
we have smooth torsion free connections on each Ui which are compatible with g.
By uniqueness, further, the connections corresponding to Ui and Uj coincide on
Ui \ Uj . Thus, these connections together define a torsion free affine connection on
M compatible with g. t
u
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5 Riemann surfaces and surfaces with Riemann metric 381
Theorem. Let M be a Riemann manifold, and let x 2 M . Then there exists an open
neighborhood of x on which R D 0 if and only if there exists an open neighborhood
U of x and a smooth map h W U ! Rn which is an isometry onto its image.
Proof. The necessity of 0 curvature for the existence of h follows directly from
Theorem 3.4, and the sufficiency almost does. In effect, if curvature vanishes in a
neighborhood of x, from Theorem 3.4, we get an open neighborhood U of x and
a map h W U ! Rn which is a diffeomorphism onto its image such that h maps
the Levi-Civita connection on U to the Euclidean connection on hŒU . Clearly,
we may then assume that U D M and h is the identity. Note however that we
have not proved the map h preserves Riemann metrics. In effect, we must investi-
gate the question: What Riemann metrics is the Euclidean connection r compatible
with?
To answer this question, assume, without loss of generality, that U is connected
(in fact, we could assume without loss of generality that it is an open ball). We
see from the formulation (4.2.1) of compatibility of the connection with the metric
that given an inner product gx on TM x for a chosen point x 2 U , there is at most
one Riemann metric gij on U with which r is compatible and such that .g ij /x D
gx (since the inner product on TM y for all y 2 U is then determined by parallel
transport). Since, however, for the Euclidean connection, parallel transport is simply
the identity when we make the canonical identification of TM y with Rn , for any
inner product gx on TM x D Rn , there is precisely one Riemann metric with which r
is compatible, namely the one specified by the same inner product on all TM y D Rn .
Since any two inner product spaces of the same dimension are isomorphic, to get
the desired isometry, it suffices to pick an affine map ˛ W Rn ! Rn which takes the
inner product on TM x to the standard inner product on Rn for a single point x 2 U .
We may then put h D ˛jU . t
u
Despite the fact that both concepts are attributed to Riemann, a Riemann surface is
not the same thing as a Riemann manifold which is a surface (i.e. has dimension 2).
A Riemann surface † is of course, in particular, a 2-dimensional manifold, and
hence Lemma 4.1.1 applies. Additionally, † comes with the structure of a complex
manifold, but that is not the same thing as a Riemann metric.
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382 15 Tensor Calculus and Riemannian Geometry
We conclude that
Q yQ D jj2 dxdy:
dxd (5.1.3)
!i D h!j
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5 Riemann surfaces and surfaces with Riemann metric 383
where h is a positive smooth real function. Thus, if ui is, again, a smooth partition
of unity subordinate to .Ui /, then
X
!D ui !i
i
This is our first encounter with the equation of holomorphic disks. In order to
solve the equation, however, it is more convenient to write it in terms of complex
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384 15 Tensor Calculus and Riemannian Geometry
˛.z/.Jz v/ D i ˛.z/.v/:
(Note that a 1-form of type .1; 0/ with respect to the standard complex structure i is
simply of the form
.z/dz;
dz D ˛ C .z/˛
dz D ˛ C ˛;
and hence
dz dz
˛D :
1 jj2
Thus, the complex 1-form dz .z/dz is of J -type .1; 0/ and the condition of f
being J -holomorphic means that
@f @f
f .dz/ D dz C dz;
@z @z
@f @f
f .dz/ D .f / .dz/ D dz C dz:
@z @z
Thus, we have
@f @f @f @f
f .dz .z/dz/ D . .f .z// /dz C . .f .z// /dz:
@z @z @z @z
The condition that this be a form of type .1; 0/ with respect to the standard complex
structure then reads
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5 Riemann surfaces and surfaces with Riemann metric 385
Our goal is then to solve the differential equation (5.2.2). To this end, we will
make one more reduction. Applying @=@z to (5.2.2) and writing
@ @
g.z/ D ; h.z/ D ; (*)
@z @z
we obtain
we therefore have
Putting
b.z/ C .z/b.z/
a.z/ D ; (5.2.4)
1 j.z/j2
this gives the equation
Our strategy is first to solve the equation (5.2.5), and then show that the solution
(with suitable conditions) also satisfies (5.2.2), and hence (5.2.1).
Before doing so, however, let us briefly consider what restriction we can place
on the function a.z/. Note that this function is related to the smooth function .z/
by the equations (*), (5.2.3) and (5.2.4). On the function .z/ we can certainly
impose the relation
.0/ D 0;
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386 15 Tensor Calculus and Riemannian Geometry
since we are free to choose the differential of f to preserve the complex structure
at 0. Further, by substituting t D ız for ı > 0 small if necessary, we can make
.z/ and its first several chosen partial derivatives arbitrarily small in a chosen
neighborhood of 0, and further, since we are only interested in a correct solution in
a neighborhood of 0, we may assume .z/ D 0. for jzj > 1=2. Using the equations
(*), (5.2.3) and (5.2.4), we can translate this to similar conditions on a.z/, i.e., for
any fixed chosen ı > 0, we can assume
5.3 Theorem. There exists an ı > 0 such that for a smooth function a.z/ satisfying
(5.2.6), there exists a solution f .z/ to the equation (5.2.5) with @f =@z, @f =@z
continuous, f .0/ D 0,
@f
lim D 0: (5.3.2)
z!1 @z
Proof. Recall Section 5.2 of Chapter 13. We will find a solution of the form
Define
@
D A./: (5.3.4)
@z
In effect, we will solve the equation (5.3.4) in the set Q" of continuous bounded
functions on C which satisfy
"
j.z/j (5.3.5)
1 C jzj
with the metric induced from the metric on the space C.C/ of bounded continuous
functions on C (the supremum metric). Note that obviously, Q" is a closed subset
of C.C/.
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5 Riemann surfaces and surfaces with Riemann metric 387
The parameter " > 0 will be chosen later, but note that (5.3.5) implies
Q L3 .C/:
Since
where
Z 1=3
dxdy
KD ;
C .1 C jzj/3
so
Let us also assume 0 < " < 1. Now by choosing ı > 0 sufficiently small, we may
assume
and
1
jA./ A. /j j j (5.3.7)
2
D lim n :
n!1
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388 15 Tensor Calculus and Riemannian Geometry
for a suitable constant K. By Lemma 5.3.1 (2) of Chapter 13 again, there exist
constants L; > 0 such that
@P1 ..z//
D .z/ .0/
@z
@f
D .z/ C 1 .0/;
@z
j.0/j ":
@P1 ./
lim D 0: (5.3.9)
z!1 @z
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5 Riemann surfaces and surfaces with Riemann metric 389
Z Z
1 dudv 1 dsdt
@.z C /=@z D @./=@ : (5.3.10)
C D z
Note that the integrand on the right-hand side 0 outside D, which lets us restrict the
integration from C to D. This also implies that taking derivatives after the integral
sign is legal by Theorem 5.2 of Chapter 5. Now the right-hand side of (5.3.10)
obviously tends to 0 with z ! 1, which proves (5.3.9). t
u
5.4 Proposition. Any solution f .z/ of the equation (5.2.5) which satisfies the
conditions of Theorem 5.3 is also a solution of the equation (5.2.2).
@ @.f .z//
.@f =@z .f .z//@f =@z/ C @f =@z
@z @z
@f @f
D @f =@z g.f .z// C .f .z// h.f .z// :
@z @z
@
.@f =@z .f .z//@f =@z/
@z
D @f =@z h.f .z// @f =@z .f .z// .@f =@z/ :
Setting
we therefore have
@F
D A.z/ F .z/
@z
where
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390 15 Tensor Calculus and Riemannian Geometry
lim F .z/ D 0
z!1
(by (5.3.1) and the fact that has compact support). Hence, F .z/ D 0 for all z 2 C
by Theorem 5.3 of Chapter 13, which proves our statement. t
u
5.5 Theorem. Every oriented smooth surface † with a Riemann metric has a
compatible complex structure. u
t
Note that in view of the comments of Subsection 5.3 of Chapter 10 and the
Riemann Mapping Theorem 1.2 of Chapter 13, this can be equivalently phrased
to say that for every surface † with a Riemann metric, and any point x 2 †,
any sufficiently small simply connected open neighborhood of x can be mapped
conformally bijectively onto
.0; 1/. In cartography, this theorem is of major
significance: Note that together with the Riemann Mapping Theorem, we can make
a flat local chart of any (smooth) landscape in the shape of any simply connected
open set in C (other than C itself) which preserves surface angles.
6 Exercises
(1) Let M be a smooth manifold with an affine connection and let U be an open
subset of M . Let x i , y i be two different coordinate systems on U , and let
k
ijk be the Christoffel symbols with respect to the coordinates x i , and ij the
Christoffel symbols with respect to y i . Prove that
k @x p @x q @y k r @y k @2 x m @2 x m
ij D pq C :
@y i @y j @x r @x m @y i @y i @y j
Note that the second term is the “error term for the symbol ijk behaving as a
tensor of type .2; 1/”.
(2) Prove that the inverse of a positive-definite symmetric matrix is positive-
definite. [Hint: We have x T Ax > 0 when x ¤ 0, and we want to prove
y T .A1 /y > 0 for y ¤ 0. Consider y D Ax.]
(3) Let M be a Riemann manifold with Riemann metric g. Define, for x; y 2 M ,
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6 Exercises 391
manifold. [Hint: Use Theorem 4.3.2 of Chapter 14. Keep in mind that one of
the things to show is that .x; y/ D 0 implies x D y.]
(4) Prove that the conditions (4.2.1) and (2) of 4.2 are equivalent. [Hint: Integrat-
ing condition (2) along a curve where r 0 .v/ D r 0 .w/ D 0, u D 0
gives (4.2.1). This also means that (4.2.1) implies (2) at points where ru .v/ D
ru .w/ D 0. Fixing local coordinates, the general case then follows by the
chain rule.]
(5) Volume associated with a Riemann metric:
(a) Let g be a Riemann metric defined on a bounded open subest U Rn .
Assuming B U is a Borel set, define
Z q
volg .B/ D det.gij /:
B
Prove that volg .B/ does not depend on the choices (i.e. the atlas and the
set Bp ).
(6) Let W ha; bi ! .0; 1/ be a smooth function (taking one-sided derivatives at
the boundary points). Consider the smooth map of manifolds
W .a; b/
S 1 ! R3
given by
S 2 D f.x; y; z/ 2 R3 j x 2 C y 2 C z2 D 1g
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392 15 Tensor Calculus and Riemannian Geometry
with the Riemann metric induced from R3 . State precisely and prove
that geodesics are precisely segments of great circles parametrized by arc
length.
(b) Generalize this to the n-sphere.
(c) Construct a Riemann metric on R2 in which there exists a geodesic with
boundary points A, B which does not minimize the distance functional
among continuously differentiable curves with boundary points A, B.
[Hint: Remove a point from S 2 , and induce a Riemann metric on R2 from
the Riemann metric (a) via the radial projection diffeomorphism.]
(8) Let M N be a smooth submanifold, and let g be the Riemann metric on
M induced by a Riemann metric gQ on N . If we denote by r resp. rQ the
Levi-Civita connection of g resp. g, Q prove that .ru .v//x is the g-orthogonal
Q
projection of rQ u .v/ onto TM x for x 2 M (note that rQ u .v/ is only defined
in the sense of 2.2). Use this to compute the curvature tensor of S 2 with the
Riemann metric induced from R3 . Conclude that no non-empty open set of S 2
is isometric to an open set of R2 (with the respective Riemann metrics). This
fact was first rigorously proved by Gauss.
(9) Prove that every 1-dimensional manifold is diffeomorphic either to S 1 or to
R. [Hint: Use Lemma 4.1.1 and parametrization by arc length.]
(10) Consider the ball S in R3 given by the equation
x 2 C y 2 C .z 1/2 D 1:
z D x C iy;
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1.1
In this chapter we will work with vector spaces over the field R of real numbers
and the field C of complex numbers (see Appendix A). Since the case of C is
perhaps less familiar, we will emphasize it, especially in the theory of Hiblert spaces.
All we say for C there remains true essentially verbatim over the field R as well,
and the reader is encouraged to consider what changes are appropriate in the real
case (mostly, complex conjugation disappears). In the case of Banach spaces, the
cases of R and C are sometimes really different. In those cases, we will spell out
both alternatives in detail.
Now recall the notion of an inner product from 4.2 of Appendix A and its
associated norm (and hence metric) from 1.2.3 of Chapter 2. Recall also the general
notion of a norm as introduced in 1.2 of Chapter 2.
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394 16 Banach and Hilbert Spaces: Elements of Functional Analysis
Chapter 5 says that Lp .B/ and Lp .B; C/, 1 p 1, are real resp. complex
Banach spaces. In fact, on L2 .B/, L2 .B; C/ we have a real (resp. complex) inner
product defined by
Z
f g D fg
B
which is finite by the Cauchy-Schwarz inequality applied at every point. Since the
norm on L2 is the norm corresponding to this inner product, the spaces L2 .B/ and
L2 .B; C/ are real and complex Hilbert spaces. The spaces Lp .B/, Lp .B; C/ are, in
some sense, the most fundamental examples.
Proof. We have jjxjj D jjy C .x y/jj jjyjj C jjx yjj and similarly with the roles
of x and y reversed, so
A subspace of a Banach resp. Hilbert space is a subset that is a Banach resp. Hilbert
space in the inherited structure. In particular, it is required to be complete. Thus, by
Proposition 7.3.1 of Chapter 2,
subspaces of a Banach resp. Hilbert space are precisely closed linear (vector)
subspaces.
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2 Uniformly convex Banach spaces 395
"2 1 1 1
1 < .x C y/.x C y/ D .1 C yx C xy C 1/ D .2 C yx C xy/
4 4 4 4
and consequently
xy C yx > 2 "2 ;
so
2.3 Lemma. Let yn ; zn be elements of a uniformly convex Banach space such that
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396 16 Banach and Hilbert Spaces: Elements of Functional Analysis
zn zn
jjzn jj
lim
D lim 1
.1 /zn
D 0: (2.3.1)
jjzn jj jjyn jj
jjzn jj jjyn jj
Since the norm is a continuous function, it follows from (2.3.1) and the assumptions
that
1 zn yn
1 zn C yn 1 zn zn
lim
. C
/ D lim
C . /
D 1
2 kzn k kyn k
kyn k 2 2 kzn k kyn k
yn zn
lim
D0
jjyn jj jjzn jj
yn zn zn zn
2.4 Theorem. Let K be a closed convex subset of a uniformly convex Banach space
B and let a 2 B. Then there exists precisely one element y 2 K such that
lim jjxn jj D 1:
Suppose that the sequence .xn /n is not Cauchy. Then there exist subsequences .yn /n
and .zn /n such that for some "0 > 0 and all n,
jjyn zn jj "0 :
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3 Orthogonal complements and continuous linear forms 397
However, we have lim jjyn jj D lim jjzn jj D 1 and by (2.4.1) also lim jj yn Cz
2
n
jj D 1 and
hence by Lemma 2.3, lim jjyn zn jj D 0, a contradiction.
Thus, .xn /n is a Cauchy sequence and if we set y D lim xn we have y 2 K and
jjyjj D 1. If we had jjzjj D 1 for another z 2 K we would have, according to the
same reasoning as above, a Cauchy sequence y; z; y; z; : : : ; y; z; : : : . t
u
3.1
M ? D fx j xy D 0 for all y 2 M g:
Note that from the property xx D 0 ) x D o of the scalar product it follows that
M \ M ? D fog:
x D y Cz with y 2 M and z 2 M ? :
and hence
zu uz zu zu
jjzjj2 zu uz C uu D jjzjj2 0; hence
uu uu uu uu
jzuj2 D .zu/zu D .zu/.uz/ 0
so zu D 0, and finally z 2 M ? .
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398 16 Banach and Hilbert Spaces: Elements of Functional Analysis
zz D zx zy D 0 0 D 0;
3.4
.M \ N /? D M ? C N ? and .M C N /? D M ? \ N ?
3.5 Theorem. Let V; V 0 be normed vector spaces (real or complex). Then the
following statements for a linear operator f W V ! V 0 are equivalent.
(1) f is continuous.
(2) f is uniformly continuous.
(3) There exists a number K such that
" K "
jjf .x/ f .y/jj D jjf .x y/jj D jj " f .x y/jj K D ": t
u
K K
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3 Orthogonal complements and continuous linear forms 399
3.5.1
This leads to a concept of a norm of a continuous linear map f W V ! V 0 between
normed vector spaces defined by
L.V; V 0 /
of all continuous linear maps f W V ! V 0 (with the natural addition and multipli-
cation by scalars).
3.5.2
A linear form on a real or complex normed vector space V is a continuous linear
mapping V ! R resp. V ! C. Similarly as in 1.1 of Chapter 11, we will denote by
V
the space of all linear forms on V . This is called the dual space of the normed
vector space V . Note, however, that, unlike in 1.1 of Chapter 11, we now take
the continuous linear forms only. The definition from 3.5.1 yields a norm on V
defined by
3.5.3
Similarly as in 1.2 of Chapter 11, we have for a continuous linear mapping f W V !
V 0 a linear mapping f W .V 0 / ! V defined by
f .'/ D ' ı f
(if f; ' are continuous then the composition ' ı f is continuous as well). We will
show that f is continuous. This is an immediate consequence of the following
Proof. We have jjf .'/jj D jj' ı f jj D supfj'.f .x//j j jjxjj 1g. If jjxjj 1 then
jf .x/j jjf jj. Thus, 1 jjf .x/jj 1 and 1 j'.f .x//j D j'. 1 f .x//j jj'jj.
jjf jj jjf jj jjf jj
t
u
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400 16 Banach and Hilbert Spaces: Elements of Functional Analysis
M D fx j '.x/ D 0g
is a subspace unequal to H and hence M ? ¤ fog, by 3.2. First we will show that
dim M ? D 1. Indeed, let o ¤ x; y 2 M ? and consider u D '.y/x '.x/y. Then
M ? D f˛b j ˛ 2 Cg
x D xM C ˛.x/b with xM 2 M:
Hence we have
'.b/
'.x/ D xa where a D :
bb
Proof. If jjxjj 1 then j'.x/j D jxaj jjxjjjjajj jjajj. On the other hand we have
'. 1 a/ D 1 aa D jjajj. t
u
jjajj jjajj
3.7.1
A map f between vector spaces over C is said to be antilinear if it preserves
addition and sends ˛z to ˛f .z/.
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3 Orthogonal complements and continuous linear forms 401
3.7.2 Remark
Note that in the case of Hilbert spaces over R, the mappings H are norm preserving
isomorphisms.
3.8
This calls for a closer analysis. For a continuous linear mapping f and a fixed
y 2 H 0 we have the linear form, obviously continuous,
h D .x 7! f .x/y/:
h D .x 7! xz/:
This mapping f Ad is referred to as the mapping adjoint to f . We will show that the
mapping g from the diagram above is equal to .f Ad / . Indeed, we have
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402 16 Banach and Hilbert Spaces: Elements of Functional Analysis
3.9
3.9.1
The eigenvalues of a linear operator f W H ! H are numbers such that f .u/ D
u for a non-zero u, and that the x’s satisfying such equations are called eigenvectors
(compare 5.1 of Appendix B). We have
4.1
We say that a system .xj /j 2J of elements of a Hilbert space has a sum x and write
X
xD xj
J
if for every " > 0 there exists a finite J."/ J such that for every finite K such
that J."/ K J we have
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4 Infinite sums in a Hilbert space and Hilbert bases 403
X
jjx xj jj < ":
j 2K
4.2 Theorem. .xj /j 2J has a sum if and only if for every " > 0 there exists a finite
subset K."/X J such that for each finite subset K J satisfying K \ K."/ D ;
one has jj xi jj < ".
K
Proof. ) : Consider an " > 0 and put K."/ D J. "2 /. Let K be finite and such that
K \ K."/ D ;. Then we have
X X X X X
jj xj jjDjj xj xj jjjj xj xjj C jj xj xjj < ":
K K[K."/ K."/ K\K."/ K."/
X
( : Set Kn D K.1/[K. 21 /[ [K. n1 / and yn D xj . From the assumption
j 2Kn
is a Cauchy sequence and hence it has a limit x D lim yn .
we easily see that .yn /nX
We will show that x D xj .
J
Choose an " > 0 and an n such that jjx yn jj < 2" and at the same time n1 < 2" .
Take a K Kn and set L D K X Kn . Then
X X X " "
jjx xj jj D jjx yn C xj jj jjx yn jj C jj xj jj C D ":
K L L
2 2
4.3 Theorem. A system .xj /j 2J has a sum x if and only if either J is finite and
X
xj D x in the ordinary sense, or the following conditions hold simultaneously:
j 2J
(a) for at most countably many j , xj ¤ o,
(b) whenever we order the xj ¤ o in a sequence x1 ; x2 ; : : : we have
X
n
lim xk D x
n
kD1
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404 16 Banach and Hilbert Spaces: Elements of Functional Analysis
J D f1; 2; : : : ; n; : : : g:
For " > 0 choose n" such that J."/ f1; 2; : : : ; n" g. Then for n n" , we
obviously have
X
n
jjx xk jj < ":
kD1
X
( : Suppose the sum xj does not exist. Choose a fixed order x1 ; x2 ; : : : .
J
X
n X
Then the limit x D limn xk either does not exist or it does but it is not xj .
kD1 J
In the latter case, by the definition, there exists an a > 0 such that
X
8 finite L J 9 finite K.L/such that L K.L/ J and jj xj xjj a:
K.L/
Put
A1 D f1g; B1 D K.A1 /;
A2 D f1; 2; : : : ; max B1 C 1g; B2 D K.A2 /
Now A1 B1 ¨ A2 B2 ¨ A3 and
X X
lim jj xj xjj D 0 while jj xj xjj a: (4.3.1)
n
An Bn
A1 ; B1 X A1 ; A2 X B1 ; : : : ; An X Bn1 ; Bn X An ; AnC1 X Bn ; : : :
(the xj in the individual blocks ordered arbitrarily), we see that in view of 4.3.1,
Xn
limn yk does not exist. t
u
kD1
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4 Infinite sums in a Hilbert space and Hilbert bases 405
X X
4.4 Theorem. Let xj and yj exist in a Hilbert space H . Then
X J J X
(1) ˛xj exists and is equal to ˛ xj ,
X
J J X X
(2) .xj C yj / exists and is equal to xj C yj , and
J X J J X
(3) for every z the sum .xj z/ exists and is equal to . xj /z.
J J
4.5
Proof. I. Existence:
) : Consider the sets K."/ from 4.2. If K J is finite and K \ K."/ D ;
then, using orthogonality,
X X X X X 2
jjxj jj2 D xj xk D . xj /. xj / D jj xj jj < "2 :
K j;k2K K K K
( : Reason as in the ) implication but in reverse, using, this time, the sets
K."2 /.
II. The equality:
X
Set x D xj . By 4.4(3), we have
J
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406 16 Banach and Hilbert Spaces: Elements of Functional Analysis
X X X X XX X
xx D . xj /x D .xj x/ D xj . xk / D .xj xk / D xj xj :
J J J J J J j
t
u
and hence
X
jxxj j2 jjxjj2 :
K
X
Thus, the sum jxxj j absolutely converges (recall 6.2 and 6.3 of Chapter 1). t
u
J
4.8
X
4.9 Theorem. (Parseval’s equality) One has jxxj j2 D jjxjj2 , that is the Bessel
X
J
inequality becomes equality, if and only if x D .xxj /xj .
J
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4 Infinite sums in a Hilbert space and Hilbert bases 407
Proof. Recall the beginning of the proof of Theorem 4.7: instead of the inequality
X 2 X 2
0 jjx .xxj /xj jj consider 0 D jjx .xxj /xj jj and observe that the
K K
formulas in the statement express the same fact. t
u
4.10
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408 16 Banach and Hilbert Spaces: Elements of Functional Analysis
(3))(4) : We have
X X X X
xy D . .xxj /xj /. .yxk /xk / D .xxj /.yxk /xj xk D .xxj /.yxj /:
j k j;k j
Let us now turn our attention to Banach spaces. Recall that linear maps f W V ! R,
f W V ! C for a real resp. complex vector space V are called linear forms.
5.1 Theorem. (Hahn - Banach) Let V be a real vector space and let W V ! R
be a function such that
(a) for all x; y 2 V , .x C y/ .x/ C .y/ and
(b) for every x 2 V and r 2 h0; 1/, .rx/ D r .x/.
Let V0 be a vector subspace of V and let f0 be a linear form on V0 such that
W 0 D fx C ra j x 2 W; r 2 Rg:
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5 The Hahn-Banach Theorem 409
and hence
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410 16 Banach and Hilbert Spaces: Elements of Functional Analysis
f .x/ D jf .x/j:
Thus,
jf .x/j D f .x/:
5.3
Proof. Use Theorem 5.1 resp. Corollary 5.2 with V D L, V0 D M and .x/ D
kgk kxk. t
u
5.4
Remark. Note that we speak of continuity but not of the norm: norm of f jM is
zero and would not help us.
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6 Dual Banach spaces and reflexivity 411
6.1
Proof. To fix ideas, let us consider the real case (the complex case is analogous).
Suppose .fn / is a Cauchy sequence in L . Let B be the unit ball in L. Then, by
definition, the restriction fn jB is a Cauchy sequence in the space C.B/ of bounded
continuous functions on B, which we discussed in Chapter 2 (and, in fact, the
L -distances kfm fn k are equal to the C.B/-distances). However, we already
know that the space C.B/ is complete, and thus the sequence .fn jB/ converges
uniformly to a function f0 W B ! R. Then it is immediate that the function f 2 L
defined by
6.2
Recall from Section 3.6 that for a continuous linear mapping f W L ! M , we have
a continuous linear mapping
f W M ! L by setting f ./ D f
Proposition. We have kf k D kf k.
Proof. To fix ideas, let us consider the real case (the complex case is analogous).
Choose an " > 0 and an x0 2 L such that 0 < kx0 k 1 and kf .x0 /k kf k ".
On the vector subspace frf .x0 / j r 2 Rg define a linear form g by setting
1
g.rf .x0 // D rkf .x0 /k. Then kgk D 1 (the unit ball is frf .x0 / j r g)
kf .x0 /k
and hence there is, by Proposition 5.3, a linear form 2 M such that kk 1
and .f .x0 // D kf .x0 /k. Thus, kf k kf ./k D kf k j.f0 .x0 //j D
kf .x0 /k kf k ". Since " > 0 was arbitrary we conclude that kf k D kf k. u t
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412 16 Banach and Hilbert Spaces: Elements of Functional Analysis
6.3
6.3.1 Proposition. is a linear map preserving norm, and for every continuous
linear map f W L ! M we have a commutative diagram
L
L ! L
? ?
?
fy
? :
yf
M
M ! M
Proof. Again, to fix ideas, let us work in the real case. The complex case is the
same.
Checking that is linear is straightforward. Consider the formula
By Lemma 3.6, jf .x/j kf k kxk and hence we see that k.x/k kxk.
Now fix an x ¤ o and define a linear form g W L0 D frx j r 2 Rg ! R by setting
g.rx/ D rkxk. The unit ball in L0 is the set frx j r kxk
1
g and hence kgk D 1.
By Proposition 5.3, we can extend g to a linear form f on L with kf k D 1 and we
have .x/.f / D f .x/ D kxk. Thus, k.x/k kxk.
Finally, let f W L ! M be a continuous linear map, x 2 L and 2 M .
We have
..f L /.x//./ D .f ..x//./ D .L .x/ f //./
D .x/.f .// D L .x/. f /
D .f .x// D .M .f .x///./ D ..M f /.x//./;
that is, f L D M f . t
u
6.4
6.4.1 Remark:
We have seen in Theorem 3.7.1 that the dual space of a Hilbert space H is
antilinearly isomorphic to H by the inner product. Composing the antilinear
isomorphisms
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6 Dual Banach spaces and reflexivity 413
H ! H ! .H / ;
one gets the map of 6.3, and thus a Hilbert space is always reflexive.
6.5 Proposition. Let a Banach space B not be reflexive. Then neither is the Banach
space B .
Proof. Since B is complete, the vector subspace B ŒB of B is also complete (it
is norm-isomorphic) and hence, by Proposition 7.3.1 of Chapter 2 closed in B .
By Proposition 5.4, there exists an F 2 B , a linear form on B that is non-zero
but identically zero on B ŒB. We will show that it is not in B ŒB . Suppose it is,
that is, F D B .f / for a linear form f on B. In particular, for each B .x/ we have
F .B .x// D 0. Thus,
for all x, hence f D o and finally also F D .o/ is identically zero, a contradiction.
t
u
The following construction works over R or C. To fix ideas, let us work over C. The
treatment over R is analogous. Let W be a Banach space and let W be its dual.
The weak topology of W (with respect to W ) has a basis of open sets determined
by all possible choices of elements f1 ; : : : fn 2 W , and open sets U1 ; : : : ; Un C:
The basis element corresponding to this data is
fX 2 W j X.f1 / 2 U1 ; : : : ; X.fn / 2 Un g:
6.6.1 Lemma. Let V be a normed vector space. Then the unit ball B of .V / is the
closure of the image B1 of the unit ball of V under the canonical map V ! .V / ,
with respect to the weak topology (with respect to V ).
Proof. To prove that B is contained in the closure of B1 with respect to the weak
topology, it suffices to show that every open set U in the weak topology disjoint
with B1 is also disjoint with B. For open sets U which are of the form
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414 16 Banach and Hilbert Spaces: Elements of Functional Analysis
F1 D F
and
U1 D .1 C .X.F / 1/=2; 1/
contains X but is disjoint from B1 , thus showing that X is not in the closure of B1
with respect to the weak topologiy. t
u
6.7 Theorem. (The Milman-Pettis Theorem) Every uniformly convex Banach space
V is reflexive.
Proof (The proof we present here is due to J.R. Ringrose). Let V be a uniformly
convex Banach space. By uniform convexity, for every " > 0 it is possible to choose
a ı D ı."/ > 0 such that if x; y 2 V satisfy
kxk; kyk 1; kx C yk 2 ı;
then
kx yk < ":
Now suppose V is a uniformly convex Banach space which is not reflexive. Let B
be the closed unit ball in .V / , and let B1 be the image of the closed unit ball in
V under the canonical map V ! .V / . Then B is contained in the closure of B1
under the weak topology (with respect to the space V ). Assuming B ¤ B1 , since
the canonical embedding V ! .V / is an isometry, by completeness, the image is
closed, and thus B1 is a closed subset of B. This means that there exists an " > 0
and an X 2 B such that, in .V / ,
.X; 2"/ \ B1 D ;: (*)
1
Now choose an F 2 V such that kF k D 1 and jX.F / 1j < ı where ı D ı."/.
2
Then put
1
V D fY 2 .V / j jY .F / 1j < ıg:
2
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7 The duality of Lp -spaces 415
V \ B1 Y C "B:
Since, however, the right-hand set is closed in .V / under the weak topology
(with respect to V ), while X is in the closure o V \ B1 with respect to the weak
topology (since, in that topology, V is open), we deduce that X 2 Y C "B. This is
a contradiction with (*). t
u
7.1 Theorem. For 1 < p < 1, the spaces Lp .B/, Lp .B; C/ are uniformly convex.
The remainder of this section will consist of a proof of Theorem 7.1. The first thing
we should realize is that the real and complex cases are actually somewhat different,
since in the complex case the definition of Lp uses the complex absolute value,
which, in effect, is a Hilbert space norm on C D R2 . Because of this, we don’t have
an obvious isomorphism of Lp .B; C/, considered as a real Banach space, to a real
Lp -space (although we won’t prove that they are not isomorphic). Of course, Lp .B/
is embedded into Lp .B; C/ isometrically, and hence the uniform convexity for
Lp .B; C/ implies the uniform convexity of Lp .B/. We will, however, be interested
in the opposite implication, as the proof of uniform convexity of Lp .B/, is, in fact,
somewhat simpler.
Assume, therefore, that we already know that Lp .B/ is uniformly convex, and
let .fn /, .gn / be sequences in Lp .B; C/ such that
fn C gn
kfn kp D kgn kp D 1; k kp ! 1:
2
Then certainly
k jfn j kp D k jgn j kp D 1;
and
fn C gn jfn j C jgn j
k kp k kp 1
2 2
(the second inequality by the triangle inequality), so
jfn j C jgn j
k kp ! 1;
2
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416 16 Banach and Hilbert Spaces: Elements of Functional Analysis
k jfn j jgn j k ! 0:
This means that there exist measurable functions ˛n W B ! C, j˛n .x/j D 1 for all
x, such that
kfn ˛n gn kp ! 0: (*)
From the uniform convexity of Hilbert spaces (applied to the 1-dimensional complex
Hilbert space C), we know that for each " > 0 there exists a ı > 0 such that
Denote by Sn the set of all x 2 B such that j˛n .x/ 1j > ", and denote by cn D cSn
its characteristic function (i.e. the function equal to 1 on Sn and 0 elsewhere). Then
and hence
lim kfn gn jjp lim kfn ˛n gn jjp C lim jj.1 ˛n /gn jjp
n!1 n!1 n!1
Since " > 0 was arbitrary, we are done: it suffices to prove the uniform convexity of
Lp .B/.
We will show now a simple argument proving the uniform convexity of Lp .B/
which does not generalize to the complex case, thus explaining in particular why
the reduction 7.2 pays off.
7.3.1 Lemma. Let 1 p < 1 and let f; g be non-negative real functions which
represent elements in Lp .B/. Then
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7 The duality of Lp -spaces 417
.x C y/p x p C y p :
as claimed. t
u
7.3.2 Lemma. If, in a normed vector space, sequences .xn /, .yn / satisfy
then
kxn C yn k ! 1; kxn yn k ! 1:
Proof. Using the compactness of the interval h0; 3i, by picking a subsequence, we
may assume, without loss of generality, that
kxn C yn k ! ˛; kxn yn k ! ˇ
while ˛ p C ˇ p D 2. Thus,
1 1
. .˛ C ˇ//p .˛ p C ˇ p / D 1;
2 2
and hence, since t p is a convex function on h0; 1/, ˛ D ˇ and equality occurs. u
t
fn C gn
kfn kp D kgn kp D 1; k kp ! 1:
2
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418 16 Banach and Hilbert Spaces: Elements of Functional Analysis
Put
fn C gn fn gn
xn D ; yn D :
2 2
Then
and hence
2 D .kxn C yn kp /p C .kxn yn kp /p
Z
D .jxn .t/ C yn .t/jp C jxn .t/ yn .t/jp /
B
Z
D .j jxn .t/j C jyn .t/j jp C j jxn .t/j jyn .t/j jp /
B
(Note that in the third equality, it is crucial that xn , yn are real numbers.) Now by
Lemma 7.3.2,
k jxn j C jyn j kp ! 1:
as claimed. This concludes the proof that Lp .B/ is uniformly convex, and hence,
by Subsection 7.2, the proof of Theorem 7.1. t
u
1 1
7.4 Theorem. Let B be a Borel subset of Rn . Let 1 < p < 1 and let C D1
p q
(then, of course, also 1 < q < 1). We have isometric isomorphisms of Banach
spaces
and
given by
Z
.Uq .y//.x/ D x y: (7.4.1)
B
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8 Images of Banach spaces under bounded linear maps 419
Proof. Let us prove the complex case (the real case is analogous). By Hölder’s
inequality, the integral (7.4.1) exists, and we have
Since Uq .y/ is linear, we therefore have Uq .y/ 2 .Lp .B; C// with
To deduce that Uq is an isometry, we need to show that the norms are in fact equal.
Let, therefore, y 2 Lq .B; C/ be such that kykq D 1. Let ˛ W B ! C be a
measureable function such that j˛.t/j D 1 for t 2 B and
˛.t/y.t/ D jy.t/j:
Define x.t/ D jy.t/jq=p ˛.t/. Then x 2 Lp .B; C/, and kxkp D 1. We compute:
Z Z Z
.Uq .y//.x/ D xy D jyj q=p
jyj D jyjq D 1;
B B B
8.1
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420 16 Banach and Hilbert Spaces: Elements of Functional Analysis
x C S D fx C y j y 2 S g;
S C T D fx C y j x 2 S; y 2 T g
Proof. Let .yn / be a Cauchy sequence in N . Let B be the unit ball in M . Then since
f is open, there exists a ı > 0 such that f ŒB contains all vectors of norm ı.
By passing to a subsequence, if necessary, we may assume that
1
kyn ynC1 k < :
2n
f .xn / D yn
and
1
kxn xnC1 k < :
2n ı
Then .xn / is a Cauchy sequence. Let x D lim xn . Then f .x/ D lim yn by continuity.
t
u
8.3 Lemma. Let M; M1 be normed vector spaces such that M is complete. Let
f W M ! M1 be a continuous linear map such that for each neighbourhood U
of o in M the closure of the image f ŒU is a neighborhood of o in M1 . Then for
each neighbourhood U of o the image f ŒU is a neighborhood of o (and hence f
is open).
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8 Images of Banach spaces under bounded linear maps 421
fx 2 M j kxk ˛g U:
Let
˛
Un D fx j kxk g; Vn D f ŒUn :
2n
Thus, every Vn is a neighborhood of o in M1 . We will prove that f ŒU is a
neigborhood of zero by showing that V1 f ŒU . To this end, let y 2 V1 be
arbitrary; we look for an x 2 U such that y D f .x/.
We will find inductively xk 2 Uk k D 1; 2; : : : such that for all n,
X
n
y f .xk / 2 VnC1 and
kD1
(*)
X
n
1
ky f .xk /k < :
n
kD1
y1 2 .y V2 / \ fz j ky zk < 1g \ f .U1 /;
that is, a y1 D f .x1 / with x1 2 U1 such that ky f .x1 /k < 1 and y1 D y v with
v 2 V2 , that is, y f .x1 / D v 2 V2 .
Now suppose we already have x1 ; : : : ; xn such that (*) holds. Then
X n
y f .xk / 2 f ŒUnC1 and since
kD1
X
n X
n
1
..y f .xk // VnC2 / \ fz j ky f .xk / zk < g
nC1
kD1 kD1
X
n
is a neigborhood of y f .xk / there is an xnC1 2 UnC1 such that
kD1
X
n X
nC1
.y f .xk // f .xnC1 / D y f .xk / 2 VnC2 ; and
kD1 kD1
X
n X
nC1
1
ky f .xk / f .xnC1 /k D ky f .xk /k < ;
nC1
kD1 kD1
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422 16 Banach and Hilbert Spaces: Elements of Functional Analysis
X
n
. xk /
kD1
Recall the definition of a meager set (set of the first category) from 3.3 of
Chapter 9, and the Theorem 3.4 of Chapter 9 stating that no complete space is
meager in itself (Baire’s Category Theorem).
Proof. The two alternatives exclude each other by Baire’s Category Theorem
(Theorem 3.4 of Chapter 9).
I. Suppose there is a neighbourhood U of zero such that f ŒU is nowhere dense
[1
in f ŒM . Then f is obviously not open. Furthermore, M D nU and hence
nD1
1
[
f ŒM D nf ŒU . Obviously, if A is nowhere dense, then nA is nowhere
nD1
dense also. Thus, f ŒM is meager in itself.
II. Let none of the f ŒU with U a neighbourhood of zero be nowhere dense.
Thus, each such f ŒU is a neighbourhood of some of its points. We will prove
that in fact it is a neighbourhood of o and the statement will follow from
Proposition 8.2 and Lemma 8.3.
Let U be a neighborhood of zero in M . By continuity of the addition we have a
neighborhood V 0 such that V 0 C V 0 U and by continuity of the map x 7! .x/,
V 0 is a neighborhood of zero, and finally also V D V 0 \ .V 0 / is a neighborhood
of o. The set f .V / is a neighborhood of a point y0 and since V D V 0 \ .V 0 /, it is
also a neighborhood of y0 . Consider the homeomorphism D .y 7! y y0 /.
It maps f ŒV onto f ŒV y0 and since f ŒV y0 f ŒV C f ŒV f ŒU we
have .f ŒV / f ŒU and since .y0 / D o and is a homeomorphism, f ŒU is a
neighborhood of o. t
u
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8 Images of Banach spaces under bounded linear maps 423
8.5
8.6
G D f.x; f .x// j x 2 X1 g X1
X2 :
This is a Banach space (a product of two complete metric spaces is complete). The
graph G D f.x; f .x// j x 2 M1 g is a closed vector subspace of M1
M2 and hence
it is, again, a Banach space.
Now the projection
p1 D ..x; y/ 7! x/ W G ! M1
is a continuous map. It is linear one-one and onto, and hence, by Theorem 8.5, the
inverse p11 W M1 ! G is continuous. Since also p2 D ..x; y/ 7! y/ W G ! M2 is
continuous, the composition f D p2 p11 W M1 ! M2 is continuous. t
u
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424 16 Banach and Hilbert Spaces: Elements of Functional Analysis
8.6.2 Remark:
The completeness hypothesis in Theorem 8.6.1 is essential. Consider the space
C.ha; bi/ of continuous real functions on a closed interval ha; bi with the norm
k k D maxt 2ha;bi j.t/j. Take the subspace M C.ha; bi/ consisting of the
functions with a continuous derivative (one-sided in a and b). Now M is a
normed vector space (not complete, though) and the convergence in M is uniform
convergence. By Theorem 5.3 of Chapter 1, if functions xn converge to X and if
the derivatives xn0 converge to y then x 0 exists and x 0 D y. Thus, the mapping
D D .x 7! x 0 / W M ! M of taking the derivative has a closed graph. Obviously,
however, D is not continuous; in fact it is continuous at no point x 2 M .
9 Exercises
(1) Prove that any finite-dimensional vector space V with an inner product is a
Hilbert space. Prove that the norms associated with any two inner products on
V define equivalent metrics.
(2) Prove that if f W H ! H 0 is an isometric isomorphism of Banach spaces
where H; H 0 are Hilbert spaces, then f .u/ f .v/ D u v. [Hint: there is a
formula expressing the dot product from its associated norm.]
(3) Prove that the closure of the unit ball
.o; 1/ in a Hilbert space H is compact
if and only if H is finite-dimensional.
(4) Give an example of a bounded linear operator F W H ! H , where H is a
Hilbert space, whose image is not closed.
(5) Prove that the symbol jjf jj defined in 3.5.1 is a norm on the space L.B; B 0 /
of continuous linear maps B ! B 0 for Banach spaces B; B 0 .
(6) Prove the statement of 3.4 in detail.
(7) Let V be a finite-dimensional Hilbert (Dinner product) space over C and let
f W V ! V be a Hermitian operator. Define, for x; y 2 V , B.x; y/ D f .x/y.
Prove that B is a Hermitian form.
(8) Let H; J be Hilbert spaces. A linear operator F W H ! J is called compact
if F ŒB is compact where B D fx 2 H j jjxjj 1g.
(a) Prove that if F is compact then for any bounded closed subset S H ,
F ŒS is compact.
(b) Prove that a compact operator is always bounded.
(c) An operator F W H ! J between Hilbert spaces is called finite if
its image is finite-dimensional. Prove that a finite operator is always
compact.
(d) Give an example of a compact operator between Hilbert spaces which is
not finite.
(9) Prove that if F W H ! J is a compact linear operator between Hilbert spaces,
then there exists an x 2 H such that jjxjj D 1 and jjF .x/jj D jjF jj jjxjj.
[Hint: Consider y 2 F ŒB to be of maximal norm (note that the norm is
continuous and F ŒB is compact).]
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9 Exercises 425
F .en / D sn fn (i)
and
F is 0 on the orthogonal complement of the closure of the vector
(ii)
subspace generated by e1 ; e2 ; : : : .
Prove further that the numbers sn are uniquely determined and that the
orthonormal systems .ei /, .fi / are uniquely determined up to a scalar
multiple if s1 > s2 > . The numbers si are known as singular values of
the operator F . [Hint: s1 D jjF jj. Use Exercise (9) and pass to orthogonal
complements.]
(b) Prove that
lim sn D 0: (iii)
n!1
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426 16 Banach and Hilbert Spaces: Elements of Functional Analysis
X
F G D F .ei / G.ei /:
i 2I
Prove that this is a well-defined inner product on the space HS.H; J / of all
Hilbert-Schmidt linear operators, and that moreover HS.H; J / with this inner
product is a Hilbert space.
(13) Prove that if L is a uniformly convex Banach space and 0 ¤ h 2 L , then
there exists a z 2 L such that kzk D 1 and h.z/ D khk.
[Hint: Choose a sequence zn in the unit ball of L such that h.zn / ! khk.
Uniform convexity implies that it is Cauchy.]
(14) Let B be a Borel set in Rn such that .B/ > 0. Prove that L1 .B/, L1 .B; C/,
L1 .B/, L1 .B; C/ are not uniformly convex.
[Hint: It suffices to consider the “baby” version - see Exercise (20) of
Chapter 5.]
(15) Let F W L ! M be a bounded operator where L; M are Banach spaces, and
the vector space M=F ŒL is finite-dimensional. Prove that then F ŒL is closed
in M .
[Hint: There is a finite-dimensional vector space V and an extension FQ W L ˚
V ! M which is onto, and maps V isomorphically onto M=f ŒL. Now FQ is
open and the image, under FQ , of the open subset L
V X f0g is M X F ŒL.]
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In most of this book, we worked with the Lebesgue integral which we constructed by
passing to limits from the Riemann integral. As a result, we obtained a construction
of the Lebesgue measure. At this point, however, we need to talk about measures
in greater generality. In this section, we summarize the basics of integration theory
with respect to more general measures.
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428 17 A Few Applications of Hilbert Spaces
1.1
Example: By Proposition 3.2 and Corollary 3.4.1 of Chapter 4, we know that the
Lebesgue measure on Rn can be considered as a Borel measure on Rn (if we ignore
the fact that it is defined on even more general sets).
X
n
sD ai cAi (1.2.1)
i D1
If .Ai / D 1 and ai D 0, we set the i ’th summand equal to 0. Note carefully that
a priori, the integral of s as defined may depend on the expression (1.2.1). However,
it doesn’t (see Exercise (1)). Even without knowing that fact, however, we define for
a Borel measurable function f W X ! Œ0; 1, (recall 4.4 of Chapter 4)
Z Z
f d D sup sd (1.2.3)
X X
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1 Some preliminaries: Integration by a measure 429
where the supremum is taken Rover all simple functions (1.2.1) such that s f .
Note: For a Borel set B X , B f d may be defined simply as the integral of the
restriction of f by the restriction of to B.
1.2.1 Lemma. For any Borel function f W X ! Œ0; 1 there exist simple functions
sn such that sn % f .
Proof. Put
sn .x/ D k=2n
Proof. Use Lemma 1.2.1, the Lebesgue Monotone Convergence Theorem (Theo-
rem 1.1 of Chapter 5), and recall definition 3.1 of Chapter 5. t
u
1.2.3 Theorem. (the Lebesgue Monotone Convergence Theorem for a Borel mea-
sure) Let fn % f , where fn W X ! Œ0; 1 are Borel-measurable functions. Then
Z Z
lim fn d D f d:
n!1 X X
(The second equality follows from -additivity.) Now taking the supremum of the
left-hand side of the inequality we just derived over all 0 < c < 1 and all simple
functions s f , we obtain the inequality of the statement. t
u
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430 17 A Few Applications of Hilbert Spaces
1.2.5 Comment
Let be a Borel measure on X and u W X ! Œ0; 1 a Borel-measurable function.
Then it follows from Lemma 1.2.4 and the Lebesgue Monotone Convergence
Theorem that
Z
E 7! ud
E
(This is clearly equivalent to requiring that Re.f /C , Im.f /C Re.f / and Im.f /
all have finite integrals). We then put
Z Z Z Z Z
C C
f d D Re.f / d Re.f / d C i Im.f / d Im.f / d:
X X X X X
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1 Some preliminaries: Integration by a measure 431
Proof. The second formula immediately follows from Lemma 1.2.4. To prove the
first formula, one first notes that for ˛ 0 it follows from Lemma 1.2.4, then one
checks it for ˛ D 1 and ˛ D i , and uses the second formula to pass to the case of
˛ arbitrary. t
u
Proof. We have
jfn f j 2g;
so by Fatou’s lemma (the proof of Lemma 8.5.1 of Chapter 5 works for any Borel
measure),
Z Z
2gd lim inf .2g jf fn j/d
X n!1 X
Z Z
D lim inf. 2gd jf fn j/d
n!1 X X
Z Z
D 2gd lim sup jf fn jd:
X n!1 X
R
Subtracting X 2gd from both sides,
Z
lim sup jf fn jd D 0
n!1 X
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432 17 A Few Applications of Hilbert Spaces
and hence
Z
lim jf fn jd D 0:
n!1 X
An analogue of Lemma 8.4.1 of Chapter 5 also holds by the same proof. Therefore,
Z
lim .f fn /d D 0;
n!1 X
p
2 The spaces L .X; C/ and the Radon-Nikodym Theorem
p
2.1 The spaces L .X; C/
with respect to the equivalence relation
of being equal almost everywhere (i.e.
f
g if and only if .fx 2 X j f .x/ ¤ g.x/g/ D 0/. The relation
is a
p
congruence, so L .X; C/ inherits a structure of a C-vector space from the set of
p
all functions satisfying (2.1.1). Again, elements of L .X; C/ are often (slightly
imprecisely but usually harmlessly) identified with their representative functions.
Again, we define jjf jjp to be the p’th root of the left-hand side of (2.1.1). For
p D 1, we define, again, jjf jj1 to be the infimum of M 1 such that
f .x/ M almost everywhere, and we define L1 .X; C/ to be the quotient of the
vector space of such functions by the congruence of being equal almost everywhere.
An analogue of Minkowski’s inequality (Theorem 8.2 of Chapter 5) holds by the
p
same proof, thus providing us with a norm on L .X; C/. The proof of Theorem 8.5.2
p
of Chapter 5 extends to the case of Borel measures to prove that the spaces L .X; C/
are complete, and hence are Banach spaces. In fact, all the theory of the spaces
Lp .B/, Lp .B; C/ we built up in Chapter 16 extends verbatim to the case of the
p p
spaces L .X /, L .X; C/. In particular, for 1 < p < 1, these Banach spaces
are uniformly convex and hence are reflexive; we simply didn’t want to complicate
the discussion in Chapter 16 with unnecessary generality where we didn’t need it.
(However, see Exercises 6, 7 below.)
It is worthwhile pointing out, though, that the case of Borel measures gives some
interesting examples which we haven’t seen before: Let S be a countable set with
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p
2 The spaces L .X; C/ and the Radon-Nikodym Theorem 433
p
the measure in which every element has measure 1. Then the space L .S / is
isometric to the space of all sequences .an /n2N such that
1
X
jan jp < 1
nD1
so the formula (2.1.2) defines an inner product on L2 .X; C/. Since the norm comes
from the inner product, L2 .X; C/ with the inner product (2.1.2) is a Hilbert space
(and similarly, L2 .X / is a real Hilbert space).
Proof. Consider the measure D C . We then have .X / < 1 and for every
Borel set S X , .S / .S /. Then every function in f 2 L2 .X; C/, f is
-integrable, hence -integrable. Define
Z
I.f / D f d:
X
Clearly,
I W L2 .X; C/ ! C
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434 17 A Few Applications of Hilbert Spaces
2.3
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3 Application: The Fundamental Theorem of (Lebesgue) Calculus 435
Proof. Let the ı-" condition hold. Then a set S with .S / D 0 satisfies the
hypothesis for every ı, and hence .S / < " for every " > 0.
Conversely, let be absolutely continuous with respect to . Suppose the ı-"
does not hold, i.e. there exists an " > 0 and sets Ei with .Ei / 1=2i such that
.Ei / ". Put Ai DTEi [ Ei C1 [T . Then .Ai / ", Ai Ai C1 , .Ai /
1=2i 1 , and hence . Ai / D 0, . Ai / " by -additivity. t
u
A function f W ha; bi ! R is called absolutely continuous if for every " > 0 there
exists a ı > 0 such that for any m-tuple of non-empty disjoint intervals hai ; bi i
ha; bi, i D 1; : : : ; m which satisfy
X
m
.bi ai / < ı;
i D1
we have
X
m
jf .bi / f .ai /j < ":
i D1
F W ha; bi ! R
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436 17 A Few Applications of Hilbert Spaces
by
Z
F .x/ D f:
ha;xi
Theorem. The function F has a derivative almost everywhere in ha; bi and we have
F 0 .x/ D f .x/ almost everywhere in ha; bi.
Proof. Recall that for every ı > 0, there exists a continuous function g W ha; bi!R
such that
Z
jf gj < ı:
ha;bi
h D f g:
Let " > 0. Let B be the set of all x 2 ha; bi for which there exists a t.x/ > 0 with
such that
Z
jhj > "t.x/:
hxt .x/;xCt .x/i
Let K be a compact subset of the open set B. Then there exist x1 ; : : : ; xN such that
[
N
.xi t.xi /; xi C t.xi // K:
i D1
Note that we may find i1 < < im such that the intervals
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3 Application: The Fundamental Theorem of (Lebesgue) Calculus 437
[
m
.xij 3t.xij /; xij C 3t.xij // K: (3.2.1)
j D1
t.x1 / t.xN /:
Then it suffices to let ij C1 be the smallest number i > ij such that .xi t.xi /; xi C
t.xi // is disjoint from .xik t.xik /; xik C t.xik // for k j . By (3.2.1), we see that
X Z Z
6X
m m
6 6ı
.K/ 6 t.xij / < jhj jhj :
j D1
" j D1 hxij t .xij /;xij Ct .xij /i " ha;bi "
6ı
.B/ :
"
Now the point is that for every " > 0 we can choose ı > 0 such that .B/ is
arbitrarily small. Let
C D fa x bj jh.x/j "g:
Clearly,
ı
.C / :
"
However, for
x 2 ha; bi X .B [ C /; (3.2.2)
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438 17 A Few Applications of Hilbert Spaces
for x as in (3.2.2). Now the sets B, C depend on ı and ", but writing B D B.ı; "/,
C D C.ı; "/, (3.2.3) holds for
\
x 2 ha; bi X .B.1=n; "/ [ C.1=n; "//;
n
which is almost everywhere. Since " was arbitrary, considering " D 1=k, k D
1; 2; : : : , we see that F 0 .x/ D f .x/ almost everywhere on ha; bi, as claimed. u
t
Let us now consider the harder direction, namely the integral of the derivative of a
function F W ha; bi ! R. By Proposition 3.2, it suffices to consider the case when
F is absolutely continuous.
F is increasing. (*)
We start with
3.3.1 Lemma. Let (*) hold and let F be absolutely continuous. Let S ha; bi
satisfy .S / D 0. Then .F ŒS / D 0.
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3 Application: The Fundamental Theorem of (Lebesgue) Calculus 439
as claimed. The fact that h is the derivative of F almost everywhere follows from
Theorem 3.2. t
u
X
N
G.x/ D sup jF .ti / F .ti 1 /j
i D1
a D t0 < < tN D x:
Proof. Let a y < x b. The supremum in the definition of G.x/ clearly will
not change if we take it only over such tuples .ti / which additionally satisfy ti D y
for some i . This shows that
X
N
G.x/ G.y/ D sup jF .ti / F .ti 1 /j (*)
i D1
y D t0 < < tN D x:
Now choose an " > 0. Then if F satisfies the condition of absolute continuity with
a particular ı > 0, (*) (applied to y D ai ; x D bi for each individual i in the
definition 3.1) shows that G satisfies the condition of absolute continuity for the
same ı.
To show that G F and G C F are non-decreasing, note that by definition, for
a y < x b,
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440 17 A Few Applications of Hilbert Spaces
and hence
as required. t
u
F D .G C F C x/ .G C x/: t
u
In the preceding sections, we obtained strong theorems (Theorems 2.2 and 3.3)
which used the theory of Hilbert spaces in their proofs, but Hilbert spaces were
not a part of the final statements. The role of Hilbert spaces in this and the next
section is different, namely as a framework in which intuitive statements can be
easily made rigorous. Of course, much more can be said on the subjects we touch on
here, but what we say is a good example of the role the concept plays, for example,
in mathematical physics.
Theorem. Let U Rn be a an open set and let 1 p < 1. Then the set Cc .U /
(resp. Cc .U; C/) is dense in Lp .U / (resp. Lp .U; C/).
Proof. Let us prove the complex case, the real case is analogous. Let K U be a
compact set. We will first prove that in Lp .U; C/,
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4 Fourier series and the discrete Fourier transformation 441
cK 2 Cc .U; C/ (4.1.1)
Finally, recall from Exercise (8) of Chapter 5 that for every measurable set S
U with .S / < 1, there exists an F -set K S , .S XK/ D 0, so in Lp .U; C/; cS
is in the closure of Cc .U; C/. Consequently, so is any non-negative simple function
s with finite integral (which is equivalent to s p having a finite integral). Now for any
f 0, f 2 Lp .U; C/, there are non-negative simple functions sn with sn % f .
Then
4.1.2 Comments
1. Note that unlike our previous results on Lp , Theorem 4.1.1 does not readily
generalize to an arbitrary Borel measure.
2. Also note that Cc .U / is certainly not dense in L1 .U /. Since the complement of
a measure 0 set in U is necessarily dense, on Cc .U /, L1 -convergence is uniform
convergence, and thus the closure of Cc .U / in L1 .U / consists, in particular, of
continuous functions.
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442 17 A Few Applications of Hilbert Spaces
1
p e inx ; n 2 Z (4.1.2)
2
S 1 D fz 2 C j jzj D 1g
with the topology induced by C. Now consider the R-vector subspace C.S 1 ; R/
spanned by the functions zn C zn , i.zn zn /, n 2 Z. Then is closed under
multiplication, contains a non-zero constant function and separates points, and
hence satisfies the hypotheses of the Stone-Weierstrass Theorem 6.4.1 of Chapter 9.
Consequently, every continuous function f W S 1 ! R is a uniform limit of a
sequence of elements of . Composing with the map e ix , we see that in particular,
every continuous function g W .0; 2 / ! R with compact support is a uniform
limit of functions gn which are finite linear combinations of the functions sin.nx/,
cos.nx/, n 2 Z. Therefore, every continuous function g W .0; 2 / ! C with
compact support is a uniform limit of functions gn where each gn is a finite linear
combination of the functions e inx , n 2 Z. By the Lebesgue Dominated Convergence
Theorem, a sequence in L2 .h0; 2 i; C/ which converges uniformly converges in L2 .
Since the functions gn are (finite) linear combinations of the elements (4.1.2), g is
in the closure of the subspace spanned by (4.1.2). Thus, our statement follows from
Theorem 4.1.1. t
u
4.1.4
As already remarked in Section 2.1 above, sometimes one denotes by `2 .C/ the
space L2 .Z; C/ where is the counting measure on Z, i.e. .S / is the number of
elements of S when S is finite, and .S / D 1 for S infinite. Then the assignment
X a.n/
p e inx 7! .a W Z ! C/ (*)
n2Z
2
defines an isomorphism
L2 .h0; 2 i; C/ ! `2 .C/
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5 The continuous Fourier transformation 443
in other spaces than L2 . Note, however, that by Theorem 4.9 of Chapter 16, we have
an expression for the coefficients an :
Z
1
an D p f .x/e inx : (**)
2 h0;2 i
While Exercise (15) of the previous Section gives a basis of L2 .R; C/, one may
ask if there is a more compelling analogue of formula (**) which would apply
to L2 .R; C/. There is a surprisingly simple answer, namely to apply (**) for a
continuous parameter instead of n 2 Z, and integrate over all of R, thus obtaining,
again, a function on R: Define for a function f W R ! C and for t 2 R,
Z
1
fO.t/ D p f .x/e ixt dx: (5.1.1)
2 R
(The integral on the right-hand side is the Lebesgue integral; we include the symbol
dx to emphasize that we are integrating in the variable x.)
Despite the simplicity of the generalization, it is immediately visible that
the situation will be more complicated than in the case of the discrete Fourier
transformation. For example, we cannot expect the formula (5.1.1) to work for every
f 2 L2 .R; C/: in order for (5.1.1) to make sense, f must be integrable. Conversely,
suppose (5.1.1) does make sense. Do we have fO 2 L2 .R; C/?
We will answer these questions partially: We will apply the continuous Fourier
transform formula (5.1.1) to certain subspace of functions called “rapidly decreasing
functions”, and extend it to an isometric isomorphism of Hilbert spaces
Š
F W L2 .R; C/ L2 .R; C/:
Again, much deeper and more specific convergence theorems exist, but we will not
discuss them in this text.
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444 17 A Few Applications of Hilbert Spaces
Proof. When tn ! t, then f .x/e itn x ! f .x/e itx , while jf .x/e itn x j D jf .x/j.
Thus, fO.tn / ! fO.t/ by the Lebesgue Dominated Convergence Theorem. This
proves continuity.
To prove the limit formula, first consider the case when f D c.a;bi , a < b:
we have
ˇZ ˇ
ˇ ˇ 1
ˇ e dx ˇˇ D je itb e ita j
itx
ˇ jtj
.a;bi
and the right-hand side goes to 0 with jtj ! 1. By a step function we shall now
mean a (finite) C-linear combination of the functions c.a;bi (with varying a < b).
Then we claim that for every integrable function f W R ! C and every " > 0, there
exists a step function s such that
Z
jf sj < ":
R
First, this is true for continuous functions with compact supports (by the conver-
gence of the Riemann integral). Then it is true for non-negative functions in Zdn and
hence for all integrable functions by the Lebesgue Monotone Convergence Theorem
and linearity of integrals. But
Z
jfO.t/ sO.t/j jf sj < ";
R
and thus the limit formula for s implies the limit formula for f . t
u
5.3 Lemma. Let f W R ! C be such that both f .x/ and x f .x/ are integrable.
Then fO.t/ is differentiable, and
dfO
dt
2
D ixf .x/.t/:
(Note: By the right-hand side, we mean the Fourier transform of ixf .x/, which is
a function of t.)
5.4 Lemma. Let f W R ! C have a continuous derivative, and assume f .x/ and
f 0 .x/ are integrable, and that
lim f .x/ D 0:
x!˙1
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5 The continuous Fourier transformation 445
Then we have
Proof. Compute:
Z Z
fb0 .t/ D 0
f .x/e itx
dx D lim f 0 .x/e itx dx
R a!1 ha;ai
Z Z
D lim .f .a/e ita f .a/e ita C itf .x/e itx dx/ D it f .x/e itx dx:
a!1 ha;ai R
The passages to the limit follow from the Lebesgue Dominated Convergence
Theorem. The middle equality is integration by parts (for the Riemann integral). u
t
Proof. First note that both sides of (5.5.1) make sense by the Riemann-Lebesgue
lemma, since fO and gO are continuous and bounded. Next, consider the integral
Z
f .x/g.t/e itx=a :
R2
Clearly, this integral exists (replace the integrand by jf .x/j jg.t/j), and is equal
to both sides of (5.5.1) by Fubini’s Theorem and linear substitutions x=a D u and
t=a D v. t
u
(Note that the term “rapidly decreasing” is a misnomer, since these functions are,
in fact, never decreasing.) Note that any smooth function with compact support is
rapidly decreasing (since all its derivatives will have, again, compact support). The
vector space of all rapidly decreasing functions f W R ! C is denoted by S.
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446 17 A Few Applications of Hilbert Spaces
Proof. By induction, using Lemmas 5.3 and 5.4, t m fO.n/ .t/ is a (finite) linear
4
combination of functions of the form x k f .`/ .x/.t/. Use the assumption and the
Riemann-Lebesgue lemma. u
t
Then by definition,
b
fQ D f
where x is the complex conjugate of x. It follows that the inverse Fourier transform
maps S to S.
Setting
1
g.x/ D p e x =2 ;
2
and using Exercise (15) of Chapter 5, and Exercise (18) below, (5.7.1) becomes
Z
1
f .0/ D p fO;
2 R
which is the special case of the formula we desire at the point x D 0. The general
case follows from Exercise (16) below. t
u
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5 The continuous Fourier transformation 447
Proof. We have
Z Z
b
hfO; gi
O D fOgO D f gO D
R R
Z Z
fe
gO D f g D hf; gi: t
u
R R
Then jf .t/ g.t/j < 2" for all t 2 K, while g is smooth and supp.g/ U . t
u
F ; F 1 W L2 .R; C/ ! L2 .R; C/
L2 .R; C/ ! L2 .R; C/
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448 17 A Few Applications of Hilbert Spaces
6 Exercises
(1) Prove that the expression (1.2.2) of 1.2 does not depend on the expression of a
simple function (1.2.1).
(2) Prove that the function volg .B/ on Borel subsets B of a Riemann manifold M
from Exercise (5) of Chapter 15 is a Borel measure on M .
(3) Prove that for two Riemann metrics g1 , g2 on a smooth manifold M , the Borel
measure volg1 is absolutely continuous with respect to the Borel measure volg2 .
Conclude that it makes sense to speak of a measure 0 set in a smooth manifold,
even when we do not specify a Riemann metric.
(4) Extend the Radon-Nikodym Theorem S to the case when there exist subsets
X1 ; X2 ; X such that X D Xn and .Xn / < 1. (The measure on X
is then called -finite). Note that we are keeping the assumption .X / < 1.
[Hint: Apply Theorem 2.2 for each Xn instead of X .]
(5) Prove uniqueness in the Radon-Nikodym Theorem, i.e. prove that if two
functions h1 , h2 in the statement of Theorem 2.2 satisfy the conclusion, then
they are equal almost everywhere.
(6) Prove that if B Rn is a Borel set and is a -finite Borel measure on B, then
there is an isomorphism of Banach spaces .L1 .B// Š L1 .B/, and similarly
in the complex case.
[Hint: Extend the Radon-Nikodym Theorem to a situation where instead
of the measure we have a continuous linear functional on L1 .B/ under
the condition .X / < 1 - the proof is the same! The “Radon-Nikodym
derivative” h is the function in L1 which we are seeking; Exercise (4) is also
relevant. To prove that there is a bound M such that jh.X /j < M almost
everywhere, assume for contradiction that jh.x/j > 2n on a subset Xn of
Rpositive measure, RXn disjoint. Then there exists an integrable function fn with
Xn jfn j 1=2 n
, Xn fn h D 1.]
(7) Prove that if U is an open set in Rn , then the spaces L1 .B/, L1 .B; C/ are not
reflexive.
[Hint: Use 4.1.2. Let V be a the closure of Cc .U / in L1 .B/. Prove that there
is a continuous linear form X on L1 .B/ which is 0 on Cc .U /. Consequently,
X cannot come from L1 .U /. (Consider Exercise (6).)]
(8) Prove that in Lemma 2.3, the assumption .X / < 1 is needed. Find a
counterexample and describe where the proof goes wrong when we omit this
condition.
(9) The requirement in Definition 3.1 that the intervals hai ; bi i be disjoint is
needed. Give an example showing that we get a different notion if we drop it.
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6 Exercises 449
(10) Prove that while x 2 sin.1=x 2 / has a derivative everywhere, it is not absolutely
continuous on h1; 1i, and thus the Lebesgue integral of its derivative does not
exist.
(11) Let F W ha; bi ! R be Lipschitz (see 3.1 of Chapter 6). Prove that F has a
derivative almost everywhere.
(12) In analogy of 4.1, find a Hilbert basis of the space L2 .ha; bi/ for a < b.
(13) Using 4.1, find a real orthonormal basis of the real Hilbert space
L2 .h0; 2 i; R/:
Prove that if f and g are integrable then the convolution is well defined, and
one has
1
f g D fO g:
O
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1.1
Let F be a field (in this book, it will always be either the field of reals R or the field
of complex numbers C). A vector space
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452 A Linear Algebra I: Vector Spaces
1.2 Examples
Vector spaces are ubiquitous. We present just a few examples; the reader will
certainly be able to think of many more.
1. The n-dimensional row vector space Fn . The elements of Fn are the n-tuples
.x1 ; : : : ; xn / with xi 2 F, the addition is given by
Note that F1 can be viewed as the F. However, although the operations a come
from the binary multiplication in F, their role in a vector space is different. See
5.1 below.
2. Spaces of real functions. The set F .M / of all real functions on a set M ,
with pointwise addition and multiplication by real numbers is obviously a
vector space over R. Similarly, we have the vector space C.J / of all the
continuous functions on an interval J , or e.g. the space C 1 .J / of all continuously
differentiable functions on an open interval J or the space C 1 .J / of all smooth
functions on J , i.e. functions which have all higher derivatives. There are also
analogous C-vector spaces of complex functions.
3. Let V be the set of positive reals. Define x ˚ y D xy, o D 1, and for arbitrary
˛ 2 R, ˛ x D x ˛ . Then .V; ˚; o; ˛ ./ .˛ 2 R// is a vector space (see
Exercise (1)).
We have distinguished above the elements of the vector space and the elements
of the field by using roman and greek letters. This is a good convention for a
definition, but in the row vector spaces Fn , which will play a particular role below,
it is somemewhat clumsy. Instead, we will use for an arithmetic vector a bold-faced
variant of the letter denoting the coordinates. Thus,
f D .f1 ; : : : ; fn /
for the n-tuple of functions fj W X ! R resp. C (after all, they can be viewed as
mappings f W X ! Fn ), and similarly.
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1 Vector spaces and subspaces 453
These conventions make reading about vectors much easier, and we will maintain
them as long as possible (for example in our discussion of multivariable differential
calculus in Chapter 3). The fact is, however, that in certain more advanced settings
the conventions become cumbersome or even ambiguous (for example in the context
of tensor calculus in Chapter 15), and because of this, in the later chapters of this
book we eventually abandon them, as one usually does in more advanced topics of
analysis.
We do, however, use the symbol o universally for the zero element of a general
vector space – so that in Fn we have o D .0; 0; : : : ; 0/.
1.4
1.5.1
Also the following statement is immediate.
By 1.5.1, we see that for each subset M of V there exists the smallest subspace
W V containing M , namely
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454 A Linear Algebra I: Vector Spaces
\
L.M / D fW j W subspace of V and M W g:
W1 C W2
and called the sum of W1 and W2 . (One often uses the symbol ‘˚’ instead of ‘C’
when one also has W1 \ W2 D fog.)
2.1
X
n
˛1 x1 C C ˛n xn (briefly, ˛j xj /: (*)
j D1
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2 Linear combinations, linear independence 455
X
n
to begin with; later, we will speak of a linear combination ˛j xj more loosely,
j D1
trusting that the reader will be able to tell from the context whether we will mean
the explicit formula or its result.
2.2
Proof. 1. is trivial.
2. A non-trivial linear combination demonstrating the dependence of the smaller
system demonstrates the dependence of the bigger one if we put ˛j D 0 for the
remaining summands.
3. It suffices to prove one implication, the other follows by symmetry since the first
system can be obtained from the second by using the coefficients ˇj . Thus,
Xn
let ˛1 .x1 C ˇj xj / C ˛2 x2 C C ˛n xn D o with an ˛k ¤ 0. Then we have
j D2
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456 A Linear Algebra I: Vector Spaces
2.3 Conventions
2.4 Theorem. Let M be an arbitrary subset of a vector space V . Then L.M / is the
set of all the (results of) linear combinations of finite subsystems of M .
Proof. The set of all such results of linear combinations is obviously a subspace of
V . On the other hand, a subspace W containing M has to contain all the (results of)
linear combinations of elements of M . t
u
Proof. If it is, the inclusion follows from 2.4 since L.u1 ; : : : ; un / is the smallest
subspace containing all the uj ; if we have the inclusion then the uj ’s are the desired
linear combinations, again by 2.4. t
u
is a generating set.
Proof. by induction.
X
n
If k D 1 we have v1 D ˛j uj and since v1 ¤ o by 2.2, there exists at least
j D1
one uj0 with ˛j0 ¤ 0. Now
1 X ˛j
uj0 D v1 C uj
˛j0 ˛j0
j ¤j0
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3 Basis and dimension 457
Now let the statement hold for k and let us have a linearly independent system
v1 ; : : : ; vk ; vkC1 . Then v1 ; : : : ; vk . is linearly independent and we have, after a
rearrangement of the uj ,
L.v1 ; : : : ; vk ; ukC1 ; : : : ; un / D V:
X
k X
n
vkC1 D ˛j vj C ˛j uj :
j D1 j DkC1
We cannot have all the ˛j with j > k equal to zero: since v1 ; : : : ; vk ; vkC1 are
independent, this would contradict 2.2.1 4. Thus, ˛j0 ¤ 0 for some j0 > k and
hence, first,
n k C 1;
and, second, after rearranging the uj ’s to exchange the uj0 with ukC1 we obtain
1 X k
˛j X
n
˛j
vkC1 D vj C ukC1 C uj ;
˛kC1 ˛
j D1 kC1
˛kC1
j DkC2
and hence
X k
˛j 1 X
n
˛j
ukC1 D vj C vkC1 C uj ;
˛
j D1 kC1
˛kC1 ˛kC1
j DkC2
3.1
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458 A Linear Algebra I: Vector Spaces
X
n X
n X
n
((1) is in 2.4; as for (2), if ˛j uj D ˇj uj then .˛j ˇj /uj D o and
j D1 j D1 j D1
˛j ˇj D 0; (3) is a combination of (1) and (2).)
v1 ; : : : ; vk ; ukC1 : : : ; un (*)
and this, by 1 again, has to contain a basis. But this basis cannot be a proper
subset of (*), by 2.6, since there exists an independent system u1 ; : : : ; un .
3. If u1 ; : : : ; un and v1 ; : : : ; vk are bases then by 2.6, k n and n k. t
u
3.3
The common cardinality of all bases of a finitely generated vector space V is called
the dimension of V and denoted by
dim V:
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3 Basis and dimension 459
Proof. We just have to show that W is finitely generated; the other statements are
consequences of the already proved facts (since a basis of W is a linearly indepen-
dent system in V ). Suppose W is not finitely generated. Then, first, it contains a
non-zero element u1 . Suppose we have already found a linearly independent system
u1 ; : : : ; un . Since V ¤ L.u1 ; : : : ; un / there exists a unC1 2 V X L.u1 ; : : : ; un /. Then,
by 2.2.1 4, u1 ; : : : ; un ; unC1 is linearly independent, and we can construct inductively
an arbitrarily large independent system, contradicting 2.6. t
u
3.5 Remark
We have learned that every finitely generated vector space has a basis. In fact, one
can easily prove, using Zorn’s lemma, that every vector space has one. Indeed, let
fIj j j 2 J g
S
be a chain of independent subsets of V . Then I D fIj j j 2 J g is an independent
set again, since any finite subset M D fx1 ; : : : ; xn g I is independent: if xk 2 Ijk
then M Ir , the largest of the Ijk , k D 1; : : : ; n. Thus there exists a maximal
independent set B and this B is a basis: if there were x … L.B/ we would have
fxg [ B independent, by 2.2.1 4, contradicting the maximality.
Recall the sum of subspaces from 1.7. We have
u1 ; : : : ; uk ; vkC1 ; : : : ; vr of W1 ; and
u1 ; : : : ; uk ; wkC1 ; : : : ; ws of W2 :
u1 ; : : : ; uk ; vkC1 ; : : : ; vr ; wkC1 ; : : : ; ws
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460 A Linear Algebra I: Vector Spaces
obviously generates W1 C W2 and hence our statement will follow if we prove that it
is linearly independent (and hence a basis) – since then dim.W1 C W2 / D r C s k.
To this end, let
X
k X
r X
s
˛j uj C ˇj vj C j wj D o:
j D1 j DkC1 j DkC1
Then we have
X
r X
k X
s
ˇj vj D ˛j uj j wj 2 W1 \ W2
j DkC1 j D1 j DkC1
X
k
and since it also can be written as ıj uj , all the ˇj are zero, by 3.1.1.
j D1
Consequently,
X
k X
s
˛j uj C j wj D o
j D1 j DkC1
and since u1 ; : : : ; uk ; wkC1 ; : : : ; ws is a basis, also all the ˛i and i are zero. t
u
4.1
In this section, it is important that we work with vector spaces over R or C. Since
all the formulas in the real context will be special cases of the respective complex
ones, the proofs will be done in C.
Recall the complex conjugate z D z1 i z2 ofp z D z1 C i z2 , the formulas z C z0 D
z C z0 and z z0 D z z0 , the absolute value jzj D zz, and realize that for a real z this
absolute value is the standard one.
4.2
..x; y/ 7! x y/ W V
V ! C resp. R
such that
(1) u u 0 (in particular always real), and u u D 0 only if u D o,
(2) u v D v u (u v D v u in the real case),
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4 Inner products and orthogonality 461
.v C w/u D vu C wu:
Remark: The notation for an inner product sometimes varies. The most common
alternate notation to x y is hx; yi (although one must beware of possible confusion
with our notation for closed intervals). The notation is particularly convenient when
we want to express the dependence of the product on some other data, such as a
matrix (see Section 7.7 below).
Further, we introduce the norm
p
jjujj D uu:
In the row vector space we will use without further mentioning the inner product the
symbol
X
n X
n
xy D xj y j (in the real case x y D xj yj /
j D1 j D1
(see Exercise (2)). This specific example of an inner product is sometimes referred
to as the dot product.
p p
4.4 Theorem. (The Cauchy-Schwarz inequality) We have jxyj xx yy.
Proof. We have
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462 A Linear Algebra I: Vector Spaces
4.5
L.v1 ; : : : ; vk / D L.u1 ; : : : ; uk /:
X
k
w D ukC1 .ukC1 vj /vj :
j D1
X
k
wvr D ukC1 vr .ukC1 vj /.vj vr / D ukC1 vr ukC1 vr D 0:
j D1
X
k
We have w ¤ o since otherwise ukC1 D .ukC1 vj /vj 2 L.v1 ; : : : ; vk / D
j D1
L.u1 ; : : : ; uk / contradicting the linear independence of u1 ; : : : ; uk ; ukC1 . Thus we
can set
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4 Inner products and orthogonality 463
w
vkC1 D
jjwjj
by 2.5.
Finally observe that if u1 ; : : : ; ur was already orthonormal, the procedure yields
vj D uj until j D r. t
u
4.6
W ? D fu 2 V j uv D 0 for all v 2 W g:
W1 W2 ) W2? W1? :
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464 A Linear Algebra I: Vector Spaces
5 Linear mappings
5.1
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5 Linear mappings 465
We have an immediate
5.2 Examples
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466 A Linear Algebra I: Vector Spaces
In view of Theorem 5.5, it is an interesting question if for any set S , we can find
Š
a vector space with a basis B and a bijection W S !B. This is called the free
F-vector space on the set S , and denoted by FS (it is customary to treat as the
identity, which is usually OK, since it is specified). Of course, for S finite, we may
simply take Fn where n is the cardinality of S . However, for S infinite, the Cartesian
product FS turns out not to be the right construction. Rather, we set
there exists a finite subset F S such that
FS D a W S ! F j :
a.s/ D 0 for s 2 S X F
x0 C W D fx0 C w j w 2 W g
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5 Linear mappings 467
L D x0 C W
is uniquely determined, while for x0 one can take an arbitrary element of L. The
space W is sometimes referred to as the associated vector subspace of V , and the
dimension of V is referred to as the dimension of L.
Proof. We have
f WL!M
f .x/ D y0 C g.x x0 /
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468 A Linear Algebra I: Vector Spaces
6.1
V =E;
and that
pE D .x 7! Œx/ W V ! V =E
x C WE :
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7 Matrices and linear mappings 469
6.2.1
If W is a subspace of V we will use, in view of 5.2, the symbol
V =W instead of V =EW :
We call the vector space V =W the quotient space (or factor) of V by the
subspace W .
6.3
Kerf:
Theorem. (The homomorphism theorem for vector spaces) For every linear map-
ping f W V ! Z and every subspace W Kerf there is an homomorphism
h W V =W ! Z
In this section we will deal with vector spaces over the field of complex or real
numbers. A matrix of the type m
n is an array
0 1
a11 ; : : : ; a1n
A D @ ::: ::: ::: A
am1 ; : : : ; amn
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470 A Linear Algebra I: Vector Spaces
where the entries ajk are numbers, real or complex, according to the context. If m
and n are obvious we often write simply
.aj1 ; : : : ; aj n /; j D 1; : : : ; m
.a1k ; : : : ; amk /; k D 1; : : : ; n
is called the transposed matrix of A. There is a variant of this construction over the
field C: If A is a matrix over C, we denote by A the complex conjugate of AT ,
i.e. the matrix obtained from AT by replacing every entry by its complex conjugate.
This is sometimes called the adjoint matrix of A. A (necessarily square) matrix A
which satisfies AT D A (resp. A D A) is called symmetric (resp. Hermitian).
Multiplication. Let A D .ajk /jk be an m
n matrix and let B D .bjk /jk be an
n
p matrix. The product of A and B is the matrix
X
n
AB D .cjk /jk where cjk D ajr brk :
rD1
We obviously have
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7 Matrices and linear mappings 471
The motivation for the definition of the product will be apparent in 7.6 below,
where we will also learn more about its properties.
Clearly, all column vectors of a given dimension n also form a vector space over
F, known as the n-dimensional column vector space and denoted as Fn . We will
see that in spite of the fact that it is more convenient to write rows than columns,
the space of columns is more convenient in the sense that for columns, composition
of linear maps corresponds to multiplication of matrices without reversing orders
(see Theorem 7.6 below). Because of this, nearly all courses in linear algebra now
use the space of column vectors and not row vectors as the default model of an
n-dimensional vector space. We will follow this convention in this text as well. In
particular, we will extend the convention 1.3 to column vectors.
e1 ; : : : ; en where ei D .ei /T
(this notation conforms with 1.3; of course .ej /k D ıjk from 7.2).
The ej ’s from Fm and Fn with m ¤ n differ (and similarly for ej ), but this rarely
causes confusion. In the rare cases where it can we will display the dimension n as
n ej , n e .
j
Obviously we have
X
n
xD xj ej : (7.4.1)
j D1
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472 A Linear Algebra I: Vector Spaces
and a mapping
A 7! fA
resp.
A 7! f A
Proof. We will prove the statement about row spaces. The statement for column
spaces is analogous (see Exercise (10)). The linearity of the formula is an immediate
consequence of the definition of a product of matrices.
We have
X
n
.ej A/1k D ejr ark D ajk (*)
rD1
X
n
f .m ej / D ajk .n ek /
kD1
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7 Matrices and linear mappings 473
fI D id; fAB D fB ı fA ;
and
f I D id; f AB D f A ı f B :
Proof. We will only prove the statement for row vectors. The statement for column
vectors is analogous (see Exercise (11)). The first formula is obvious. Now let A,
B be matrices of types m
n resp. n
p. If two linear maps agree on a basis they
obviously coincide. We have
X X
fB .fA .m ej // D fB . ajk .m ek // D ajk fB .m ek /
k k
X X XX
D ajk . bkr .p er / D . ajk bkr /p er D fAB .m ej /: t
u
k r r k
7.6.1
From the associativity of composition of mappings and from the uniqueness of the
matrix in the representation of linear mappings as fA we immediately obtain
B;C f
A
WV !W
given by
X
m
B;C f .vj / D
A
aij wi :
i D1
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474 A Linear Algebra I: Vector Spaces
map (or simply matrix of the linear map) f D B;C f A with respect to the bases
B; C . An analogue of Theorem 7.6 of course holds, i.e.
B;D f
A1 A2
D C;D f A1 ı B;C f A2 (*)
for an m
n matrix A1 and an n
p matrix A2 , and ordered bases B; C; D of
m- resp. n- resp. p-dimensional spaces U , V , W .
For two ordered bases B; B 0 of the same finite-dimensional vector space V , the
matrix of Id W V ! V with respect to the basis B in the domain and B 0 in the
codomain is sometimes referred to as the base change matrix from the basis B to
the basis B 0 . By (*), base change matrices can be used to relate matrices of linear
maps with respect to different bases, both in the domain and codomain.
B.x; y/ D y Ax:
hx; yiB :
(In the real case, of course, y D yT .) Conversely, the axioms immediately imply
that every Hermitian (resp. symmetric bilinear) form on Cn (resp. Rn ) arises in this
way. We will say that the form B is associated with the matrix A and vice versa.
Sometimes we simplify the terminology and call a Hermitian (resp. real symmetric)
matrix positive definite resp. negative definite resp. indefinite if the corresponding
property holds for its associated Hermitian (resp. symmetric bilinear) form.
8 Exercises
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8 Exercises 475
(5) Prove that if 1 , 2 are inner products on a (real or complex) vector space V ,
and ; > 0, then .1 / C .2 / is an inner product.
(6) Prove that linear maps F ! F are precisely the mappings .x 7! ax/ where
a 2 F is fixed. Z b
(7) Prove that if ha; bi is a closed interval then . 7! .x/dx/ is a linear
a
mapping C.ha; bi/ ! R1 .
(8) Prove that the set of all as , s 2 S in 5.6 forms a basis of the free vector space
FS on a set S .
(9) Prove that an affine map f W L ! M between affine subsets of vector spaces
V , W can be made to satisfy the definition 5.9 with any choice of the element
x0 2 L. Is an analogous statement true for y0 2 M ?
(10) Prove the statement of Theorem 7.5.1 for column vectors.
(11) Prove the statement of Theorem 7.6 for column vectors.
(12) Prove that the set of all matrices of type m
n with entries in F is a vector
space over F where addition is addition of matrices, and multiplication by a
scalar 2 F is the operation which multiplies each entry by . Is this vector
space finite-dimensional? What is its dimension?
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1.1.1 Observation. An elementary row (resp. column) operation does not change
the row resp. column space.
1.2
The column space is, of course, changed by a row operation (and the row space is
changed by a column operation). We have, however, the following
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478 B Linear Algebra II: More about Matrices
a .x1 ; x2 ; : : : ; xn / D .ax1 ; x2 ; : : : ; xn /:
X
n
.x1 ; x2 ; : : : ; xn / D .x1 C bj xj ; x2 ; : : : ; xn /;
j D2
X
n
.x1 bj xj ; x2 ; : : : ; xn /:
j D2
1.3 Theorem. For any matrix A, the dimensions of the row and column spaces
coincide.
Proof. By 1.1.1 and 1.2, the dimensions are unchanged after arbitrarily many row
and column operations.
If ajk D 0 for all j; k then both the dimensions are zero. Let there be an ajk ¤ 0.
Performing (E1), we can move the ajk to the position .1; 1/ and multiplying the
1
(now) first row by we have our matrix transformed to
ajk
0 1
1; b12 ; : : : ; b1n
B b21 ; b22 ; : : : ; b2n C
B C
@ ::: ::: ::: A:
bm1 ; bm2 ; : : : ; bmn
Now we will perform the operations (E3) subtracting the first row bj1 times from the
j -th one, and when this is finished we do the same with the columns thus obtaining
the matrix transformed to
0 1
1; 0; : : : ; 0
B .2/ .2/ C
B0; a22 ; : : : ; a2n C
B C:
@ ::::::::: A
.2/ .2/
0; am2 ; : : : ; amn
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2 Systems of linear equations 479
.2/
If all the ajk with j; k 2 are zero, the dimension of the two spaces are 1.
.2/
Otherwise choose an ajk ¤ 0, move it to the position .2; 2/ by (E1) operations
(without affecting the first row and column) and repeat the procedure as above
to obtain
0 1
1; 0; 0; : : : ; 0
B0; 1; 0; : : : ; 0 C
B C
B .3/ .3/ C
B0; 0; a33 ; : : : ; a3n C :
B C
@ ::::::::: A
.3/ .3/
0; 0; am3 ; : : : ; amn
.rC1/
After sufficiently many repetitions of the procedure we have ajk D 0 for all
j; k > r and have a matrix
0 1
1; 0; : : : ; 0; 0; : : : ; 0
B0; 1; : : : ; 0; 0; : : : ; 0C
B C
B C
B ::::::::: C
B C
B D B0; 0; : : : ; 1; 0; : : : ; 0C
B C
B0; 0; : : : ; 0; 0; : : : ; 0C
B C
@ ::::::::: A
0; 0; : : : ; 0; 0; : : : ; 0
with the first r diagonal entries 1 and all the others zero, and hence the dimensions
of both the row and the column spaces are equal to r. t
u
1.4
The common dimension of the row and column spaces is called the rank of the
matrix and denoted by
rankA:
2.1
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480 B Linear Algebra II: More about Matrices
AxT D bT :
Thus we have a linear map f W Fn ! Fm and would like to determine the set
f 1 ŒbT :
X
n
xj cj D bT :
j D1
x aj D 0 for all j D 1; : : : ; m
where is the dot product and aj D .aj1 : : : ; aj n / are the complex conjugates of
the rows of A (this approach is valid for F D R; C, which, as remarked above,
are the only contexts we are interested in).
Thus, the set of solutions of the associated homogeneous system coincides with
the orthogonal complement
L.a1 ; : : : ; am /? :
Now the dimension of L.a1 ; : : : ; am / is the same as that of the row space, that
is, equal to the rank r od A: if we perfom the procedure from Theorem A.3.2
(the Gram-Schmidt process) on the system a1 ; : : : ; am , we end up with a basis of
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2 Systems of linear equations 481
2.3
2.3.1 Theorem (Frobenius). A system of linear equations has a solution if and only
if the rank of the matrix of the system is the same as the rank of the augmented one.
(That is: if and only if the right-hand side column is in the column space of A.)
From 2.2 1 and 2.2 3, we obtain
2.3.2 Theorem. If a system of linear equations has a solution x0 , then the set of all
solutions is an affine set
x0 C W
where W is the set of all solutions of the associated homogeneous system. The
dimension of this affine set is n rankA.
By 2.3.2, to determine the set of all solutions of the system (2.1.1), it suffices to find
one of its solutions and s D n r linearly independent solutions x1 ; : : : ; xs of the
associated homogeneous system, where r D rankA. The general solution is then
X
s
x0 C ˛j xj ; ˛j 2 F arbitrary:
j D1
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482 B Linear Algebra II: More about Matrices
0 0 0 1
1; a12 ; : : : ; a1n ; b10
B a0 ; a0 ; : : : ; a0 ; b 0 C
B 21 22 2n 2 C
@ : : : : : : : : :A
0 0 0
am1 ; am2 ; : : : ; amn ; b20
0
and then subtract from the j -th rows, j D 2; : : : ; m, the aj1 multiple of the first
one. Now we have
0 0 0 1
1; a12 ; : : : ; a1n ; b10
B0; a00 ; : : : ; a00 ; b 00 C
B 22 2n 2 C :
@ : : : : : : : : :A
00
0; am2 00
; : : : ; amn ; b200
We repeat the procedure in the part of the matrix with indices 2 (during this, of
0 0
course, the a12 ; : : : ; a1n are permuted, too; again, the j2 from the aj002 k moved to the
.2; 2/ position to be remembered). After repeating the procedure r 1 times we
obtain a matrix
0 1
1; c12 ; c13 ; : : : ; c1r ; : : : ; c1n ; bQ1
B0; 1; c23 ; : : : ; c2r ; : : : ; c2n ; bQ2 C
B C
B0; 0; 1; : : : ; c ; : : : ; c ; bQ C
B 3r 3n 3C
B C
B ::: ::: ::: C
B C
B0; 0; 0 : : : 1; : : : ; crn ; bQr C
B C
B0; 0; 0 : : : 0; : : : ; 0; 0 C
B C
@ ::: ::: ::: A
0; 0; 0 : : : 0; : : : ; 0; 0
(note that because of Frobenius’ Theorem the right-hand side becomes zero after the
r-th row or else the system has no solution) corresponding to a system of equations
X
n
y0;k1 D ck1;j y0j C bQk1 :
j Dk
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2 Systems of linear equations 483
X
n
yi;k1 D ck1;j yij :
j Dk
X
n
aij xj D bj; i D 1; 2; : : : ; n
j D1
has precisely one solution: it has a solution since the augmented matrix, being of
type n
.n C 1/, cannot have a bigger rank than n; on the other hand, the dimension
of the set of solutions is n n D 0. By 1.3,
a matrix A is regular if and only if AT is regular.
2.5.1 Theorem. The following statements about a square matrix A are equivalent.
(1) A is regular.
(2) There exists a matrix U such that AU D I .
(3) There exists a matrix V such that VA D I .
(4) The matrix A has a unique inverse matrix, that is, there is a unique U such that
UA D AU D I .
Proof. (1))(2),(3): Notation from 2.2 1 and A.7.4. For each ei on the right-hand
side there is a solution xi such that
AxTi D eTi .D ei /:
X j
X j
Thus, ajk xi k D ıi , and if we set uij D xj i we have ajk uki D ıi , that is, we
k k
have a U such that AU D I . The statement (3) is obtained applying this reasoning
for AT and using X
A.7.2.
(2))(1): Let aij ujk D ıik . Fix k and set xj D ujk . Then in the notation of
j
2.2.2 we have for the columns cj of A,
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484 B Linear Algebra II: More about Matrices
X
xj cj D ek :
j
Thus, the column space contains all the ek and hence its dimension is n.
(2)&(3))(4): If AU D I and VA D I we have have V D V .AU / D .VA/U D U .
(4))(2) is trivial. u
t
Recall now our problem from A.4.7 of deciding if a Hermitian (or real symmetric
bilinear) form is positive-definite or negative-definite. Consider a Hermitian form
B on a finite-dimensional complex vector space V (the case of a real symmetric
bilinear form is analogous). Then perform the following procedure:
Start with k D 0. Suppose we have constructed vectors v1 ; : : : vk 2 V such that
B.vi ; vi / ¤ 0, B.vi ; vj / D 0 for i ¤ j . Note that the vectors vi must be linearly
independent. (In effect, suppose
X
k
ai vi D 0:
i D1
Applying B.‹; vi /, we get ai D 0.) Then, using a system of linear equations, find a
non-zero vector w 2 V such that B.vi ; w/ D 0 for all i D 1; : : : ; k. If no such w
exists, then by 2.2 3, k dim.V /, and by linear independence, equality arises, so the
vi ’s form a basis of V . In this case, if the signs of the real numbers B.vi ; vi / are all
positive (resp. negative), B is positive-definite (resp. negative-definite). Otherwise,
B is indefinite.
Suppose the vector w exists. If B.w; w/ ¤ 0, put vkC1 D w and repeat the
procedure with k replaced by k C 1. If B.w; w/ D 0, find a vector u 2 V such that
B.w; u/ ¤ 0. If no such u exists, B is degenerate. If u exists, then
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3 Determinants 485
3 Determinants
3.1
A group G is a set with a binary operation which satisfies associativity, has a unit
element e and an inverse unary operation .‹/1 . Explicitly, the axioms are
.a b/ c D a .b c/;
a e D e a;
x x 1 D x 1 x D e:
ab Dba
where is the operations. Groups satisfying this property are called commutative or
abelian. We will soon encounter examples of groups which are not abelian.
We will not develop the theory of groups at all here (and the reader is referred to
[2] and [4] for more on abstract algebra), but they do come up naturally in the context
of the determinant. In particular we will use the obvious fact that the mappings
G!G
are bijections (the first is inverse to itself, the other one to x 7! a1 x). It then
follows that if f W G ! R or C is any mapping then
X X X
f .x/ D f .x 1 / D f .ax/ (3.1.1)
x2G x2G x2G
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486 B Linear Algebra II: More about Matrices
We will be concerned with the group P .n/ of permutations of the set f1; 2; : : : ; ng,
i.e. bijections f1; 2; : : : ; ng ! f1; 2; : : : ; ng, where the operation is composition. A
permutation p 2 P .n/ will be usually encoded as a sequence
Proof. 1. By induction. The statement is obvious for n D 1; 2. Now let it hold for
P .n/ and let p be a permutation of f1; : : : ; n; nC1g. Consider the transposition
interchanging nC1 with p.nC1/ (if p.nC1/ D nC1 set D id). Now q D ıp
sends n C 1 to n C 1, hence f1; : : : ; ng to f1; : : : ; ng. The restriction q 0 of q to
f1; : : : ; ng can be written as q 0 D 10 ı ı r0 with transpositions j0 . Extending
these to transpositions j of f1; : : : ; n; n C 1g we obtain a representation
p D ı 1 ı ı r :
(# indicates the number of elements). We will prove that for any transposition the
number
j. ı p/ .p/j
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3 Determinants 487
Permutations p with sgnp D 1 (resp. sgnp D 1) are called even (resp. odd).
3.3
It is often indicated as
ˇ ˇ
ˇa11 ; : : : ; a1n ˇ
ˇ ˇ
ˇ ::: ::: ˇ:
ˇ ˇ
ˇa ; : : : ; a ˇ
n1 nn
ˇ ˇ
ˇ a; b ˇ
ˇ
Thus for instance ˇ ˇ D ad bc (and this is about the only case of a determinant
c; d ˇ
easily and transparently computed from the basic definition).
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488 B Linear Algebra II: More about Matrices
and by (3.1.1),
X
D sgn p sgn q a1;q.1/ an;q.n/ D sgn p det A: t
u
q2P .n/
3.3.2 Corollary. If there are in a matrix A two equal colums or rows then
det A D 0.
3.4 Theorem. A determinant is linear in each of its rows (resp. columns). That is,
if A is a matrix of type n
n and if Aj .x/ is obtained from A by replacing the j -th
row by x then the mapping
is linear.
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4 More about determinants 489
3.4.1 Convention
The notation Aj .x/ will be kept in the remainder of this chapter. Furthermore, we
will use the symbol Aj .xT / for the matrix in which the i -th column is replaced
by xT .
Proof.
X Let a1 ; : : : ; an be the rows of A. We have A D Ai .ai / and B D Ai .ai C
˛j aj /. By 3.2, det Ai .aj / D 0 for j ¤ i and hence
j ¤i
X X
det B D det Ai .ai C ˛j aj / D det Ai .ai /C ˛j det Ai .aj / D det A: t
u
j ¤i j ¤i
3.4.3 Proposition. Let aij D 0 for i > j . Then det A D a11 a22 ann . More
explicitly,
ˇ ˇ
ˇ a ; a ; a ; :::; a ˇ
ˇ 11 12 13 1;n1 ; a1n ˇ
ˇ 0: a : a ; : : : ; a ˇ
ˇ 22 23 2;n1 ; a2n ˇ
ˇ ˇ
ˇ 0: 0: a33 ; : : : ; a3;n1 ; a3n ˇ D a11 a22 ann :
ˇ ˇ
ˇ ::: ::: ::: ˇ
ˇ ˇ
ˇ 0; 0; 0; : : : ; 0; ann ˇ
Proof. follows again from the definition: if p ¤ Id then there is an i with i > p.i /.
t
u
Denote by A.i;j / the matrix obtained from A by deleting the i -th row and the j -th
column. The number
˛ij D .1/i Cj det A.i;j /
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490 B Linear Algebra II: More about Matrices
4.1.1
Recall the notation from 3.4.1. We have the following
X
n X
n
Theorem. det Ai .x/ D xj ˛ij and det Aj .xT / D xi ˛ij .
j D1 j D1
Proof.
P We shall treat the case of rows (the case of columns is analogous). Since
x D xj ej we have
X
det Ai .x/ D xj det Ai .ej /:
j
Now
ˇ ˇ
ˇ a1;1 ::: a1;j 1 0 a1;j C1 ::: a1;n ˇˇ
ˇ
ˇ : : : ˇˇ
ˇ ::: ::: ::: ::: ::: :::
ˇ ai 1;n ˇˇ
ˇ ai 1;1 ::: ai 1;j 1 0 ai 1;j C1 :::
ˇ ˇ
det Ai .ej / D ˇ 0 ::: 0 1 0 ::: 0 ˇ:
ˇ ˇ
ˇ ai C1;1 ::: ai C1;j 1 0 ai C1;j C1 ::: ai C1;n ˇ
ˇ ˇ
ˇ ::: ::: ::: ::: ::: ::: : : : ˇˇ
ˇ
ˇ an;1 ::: an;j 1 0 an;j C1 ::: an;n ˇ
Exchange subsequently the i -th row with the .i 1/-th one then the .i 1/-th row
with the .i 2/-th one, etc., and then similarly operating with the rows we move the
1 from the .i; j /-th to the .1; 1/-th position and obtain
ˇ ˇ
ˇ 1 o ˇ
det Ai .ej / D .1/ i Cj ˇ ˇ i Cj
det A.i;j / D .1/i Cj ˛ij :
ˇ yT A.i;j /ˇ D .1/ t
u
AxT D bT
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4 More about determinants 491
from 2.2 1. If A is a regular matrix we can multiply this formula by A1 from the
left to obtain
The sum is then by 4.1.1 equal to det Ai .b/ so that we obtain the formula (Cramer’s
Rule)
det Aj .b/
xi D :
det A
Of course computing the solutions using this formula would be much harder than
using the Gauss Elimination. It is, however, useful for theoretical purposes.
4.3.1 Lemma. Let A; B be square matrices and let C be a matrix of the form
AM A O
or as
O B M B
where O indicates a system of zero entries while the entries at M are arbitrary.
Then
Proof. It suffices to treat the first case. Transform the matrix as indicated in 3.4.4 to
obtain
0 0 0 0 0 1
a11 a12 a13 ::: a1m
B 0 0
a22 0
a23 ::: 0
a2m C
B C
B 0 0 0
a13 ::: 0
a3m M C
B C
B ::: C
B ::: ::: ::: ::: C
B 0 C
B 0 0 0 ::: amm C
B 0 0 0 0 C :
B b11 b12 b13 ::: b1n C
B C
B 0 0
b22 0
b23 ::: 0 C
b2n
B C
B O 0 0 0
b13 ::: 0 C
b3n
B C
@ ::: ::: ::: ::: ::: A
0
0 0 0 : : : bnn
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492 B Linear Algebra II: More about Matrices
If we do the first just in the first m rows and columns and then in the remaining
ones, the left upper part corresponds to the transformation of the matrix A and the
0 0 0
right lower one is the matrix B transformed. Thus we have det A D a11 a22 amm ,
0 0 0 0 0 0 0 0 0
det B D b11 b22 bnn and det C D a11 a22 amm b11 b22 bnn D det A det B. u t
To the i -th column add the a1i multiple of the .n C 1/-th column, the a2i multiple
of the .n C 2/-th column, etc. untill the ani multiple of the 2n-th column. Then the
upper left part anihilates, and the lower left part becomes AB, schematically
O In
:
AB B
Now let us exchange the i -th and .n C i /-th rows and, to compensate the change of
sign, multiply after each of these exchanges the i -th row by -1. We obtain
In O
DD
B AB
and still det C D det D. By Lemma 4.3.1, det C D det A det B and det D D
det I det AB D det AB. t
u
Proof. If A is not regular then some of the rows are linear combinations of the
others and det A D 0 by 3.4.2. If A is regular it has an inverse A1 . Thus by 3.3.2,
det A det A1 D det AA1 D det I D 1 and hence det A ¤ 0. t
u
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5 The Jordan canonical form of a matrix 493
Av D v: (5.1.1)
Now this is a system of linear equations that has a nonzero solution if and only if
rankA < n, that is, by 4.4, if and only if
A ./ D det.I A/ D 0:
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494 B Linear Algebra II: More about Matrices
Matrices of type n
n can be added by the rule
˛A D .˛aj k /j k :
An :
p.x/ D Ck x k C : : : C1 x C C0
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5 The Jordan canonical form of a matrix 495
A .A/ D O:
By Cramer’s rule,
or
Examining the highest power of which occurs in C./T q./, we see that
C./T q./ D 0;
5.4
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496 B Linear Algebra II: More about Matrices
5.5
Y
k
A .x/ D .x i /ni I
i D1
P
thus, i are the eigenvalues of A, and ni D n. Define subspaces Wi Cn ,
i D 0; : : : ; k and linear transformations
fi W Wi 1 ! Wi ; i D 1; : : : ; k
as follows.
W 0 D Cn ;
fi D .A i E/ni jWi 1 ;
Wi D fi ŒWi 1 :
By definition,
By Cayley-Hamilton’s Theorem,
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5 The Jordan canonical form of a matrix 497
Wk D 0: (2)
dim.Ker.f0 // C C dim.Ker.fk1 // D n;
hence, by (1),
X
k
dim.Ui / n:
i D1
X
k
vi D 0; vi 2 Ui ; (3)
i D1
then v1 D D vk D 0. Let
ni D minfN j .A i I /N vi D 0g:
Suppose ni0 ¤ 0. Then replacing each vector vi by vi0 D .A i0 I /vi , the vectors
vi0 still satisfy (3) in place of the vi ’s. When we make this replacement, the number
ni0 decreases by 1, while the numbers ni , i ¤ i0 , remain unchanged. After applying
this procedure finitely many times, we achieve a situation where ni1 D 1 for some
i1 , and ni D 0 for i ¤ i1 . Then (3) reads
vi1 D 0;
(We refer to this direct sum as the Jordan canonical form of the matrix A.)
Proof. We will exhibit a proof which will allow us to find the Jordan blocks and the
matrix T explicitly (assuming we already have the eigenvalues).
Fix an eigenvalue . We shall exhibit a basis of U with respect to which the
matrix of the linear transformation AjU is a direct sum of Jordan blocks. Put f D
I A. Define subspaces
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498 B Linear Algebra II: More about Matrices
of U inductively by
Um D U ;
then all the inclusions (1) are strict. Let vj1 ; : : : ; vj qj be a set of vectors in Uj
which projects to a basis of U;j =.Uj 1 C f ŒU;j C1 /, j D 1; : : : ; m (recall
A.6.2.1). Then
is by definition the desired basis. Combining these bases over for all eigenvalues ,
by Lemma 5.5.1, gives a basis with respect to which the linear transformation A is
a sum of Jordan blocks. Further, the sizes of the Jordan blocks determine and are
determined by the dimensions of the spaces Uj , which in turn depend only on the
matrix A. This implies the uniqueness statement. t
u
6 Exercises
x C 2y C 3z C 4t C u D 10;
2x C 4y C 2z C 5t C u D 8;
3x C 6y C 5z C 9t C 2u D 1:
(3) Prove that a Hermitian form over Cn (resp. symmetric bilinear form over Rn )
is non-degenerate if and only if its associated matrix is regular.
(4) Decide whether the symmetric bilinear form on R3 associated with the matrix
0 1
461
@6 8 2A
124
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6 Exercises 499
(a; b; c; d 2 C) is regular, and write down a closed formula for its inverse.
(10) Determine the Jordan canonical form of the matrix
0 1
11 03
B0 1 1 0C
ADB
@0 0
C
1 0A
00 01
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Bibliography
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Index of Symbols
Z
.a; b/ open interval, 6
f d integral by a measure, 427
ha; bi closed interval, 6
ZXb
A adjoint matrix, 470
AT transposed matrix, 470 f .x/dx the integral, 27
a
A1 inverse matrix, 483 p
` , 433
C.X/ space of bounded continuous functions, `p .C/, 433
56 fO the Fourier transform formula, 443
C r , C 1 degrees of smoothness, 289 ln.x/ natural logarithm, 30
F , Gı , Fı . . . types of Borel sets, 123 C the field of complex numbers, 6
Lp , 138 FS free vector space on a set S, 466
`
Rijk curvature tensor, 375 F field of real or complex numbers, 451
Tijk torsion tensor, 375 Fn the space of column vectors, 471
V dual vector space, 268 Fn row vector space, 452
W .y1 ; : : : ; yn / Wronskian, 180 R the field of real numbers, 4
W ? orthogonal complement, 463 Z functions with compact support on Rn , 106
Œu; v Lie bracket of vector fields, 167 Zup , Zdn , Z sets of certain limits of compactly
ƒ Lebesgue measurable functions, 118 supported functions, 107
ei , ei standard bases, 471 B Borel sets, 123
o the zero element of a vector space, 452 F Fourier transformation, 443
u v inner product, dot product, 461 F 1 inverse Fourier transformation, 447
v row or column vector, 452 S the space of rapidly decreasing functions,
A characteristic polynomial of a matrix, 493 445
j
ıi Kronecker delta, 359 L Lebesgue integrable functions, 110
det A, jAj determinant of a matrix, 487 Lup , Ldn , L functions with a (possibly
dim.VZ / dimension of a vector space, 458 infinite) Lebesgue integral, 113
TM x the tangent space at a point x, 293
.I / line integral of the first kind, 199
ZL sgn p sign of a permutation, 487
@f
.II/ line integral of the second kind, 199 partial derivative, 66
Z L @xi
@v f directional derivative, 68
f Lebesgue integral, 109 Df total differential, 73
Z
d exterior derivative, 298
f Riemann integral over an n-dimensional definition of, 31
J
Z interval, 99 1 .†; x0 / fundamental group, 338
', 325
f .z/dz complex line integral, 202
ZL sin.x/; cos.x/ trigonometric functions, 30
Col.A/ column space, 477
! integral of a differential form, 302
ZB Row.A/ row space, 477
Im.z/, 6
f Lebesgue integral over a set, 124
M Re.z/, 6
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504 Index of Symbols
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Index
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506 Index
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Index 507
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508 Index
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Index 509
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510 Index
Young’s inequality, 16
Variation of constants, 164, 181
Vector, 451
field, 166, 295 Zero, 256
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