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Concepts of basis in functional analysis

Idea

A basis in functional analysis is a linear basis that is compatible with the topology of the underlying topological vector space. Therefore this is sometimes also referred to as a “topological basis”, but beware that this term is also used for referring to the unrelated concept of a “basis for the topology”.

Bases in linear algebra are extremely useful tools for analysing problems. Using a basis, one can often rephrase a complicated abstract problem in concrete terms, perhaps even suitable for a computer to work with. A basis provides a way of describing a vector space in a way that:

1. Is complete: every point in the space can be described in this fashion.
2. Has no redundancies: the description of a point is unique.

When translated into the language of linear algebra, we recover the key properties of a basis: that it be a spanning set and linearly independent.

In infinite dimensions, having a basis becomes more valuable as the spaces get more complicated. However, the notion of a basis also becomes complex because the question of what makes a description admits different answers depending on whether we want only finite sums, we allow sequences, or we want infinite sums.

Definitions

Properties

1. In the presence of the axiom of choice, Hamel bases always exist.

2. If $B$ is a topological basis, then $B$ has a dual basis. Since $B \setminus \{b\}$ is not total but $B$ is total, the closure of the span of $B \setminus \{b\}$ must be a codimension $1$ subspace, whence the kernel of a non-trivial continuous linear functional on $V$, say $f_b$. By scaling, this functional can be assumed to satisfy $f_b(b) = 1$. Since $B \setminus \{b\} \subseteq \ker f$, $f(b') = 0$ for all $b' \in B$, $b' \ne b$.

3. If $B$ is a Schauder basis then it is a topological basis and so, as mentioned, has a dual basis. Then the coefficients in the sum $v = \sum \alpha_b b$ must be given by evaluating the dual basis on $v$: $v = \sum f_b(v) b$.

Examples

1. In $C([0,1],\mathbb{C})$ with the norm ${\|f\|} = \max\{{|f(t)|}\}$:

   1. The monomials are linearly independent and have dense span, but do not form a topological basis as there is a sequence of polynomials with no linear term converging to $t$.

   2. The trigonometric polynomials do form a topological basis. The dual basis is given by taking the Fourier coefficients of a function. However, it is not a Schauder basis as there are continuous functions which are not the uniform limit of their Fourier series.

   3. The following is a Schauder basis. Let $(d_n)$ be the sequence $\{0, 1, \frac{1}{2}, \frac{1}{4}, \frac{3}{4}, \dots\}$. Define $f_n$ to be the piecewise-linear function with the property that: $f_n(d_n) = 1$ and $f_n(d_k) = 0$ for $k \lt n$, and $f_n$ has the least “breaks”. Then $f_n$ forms a Schauder basis for $C([0,1],\mathbb{C})$. This is the classical Faber-Schader basis.

References

• Enflo, P. (1973). A counterexample to the approximation problem in Banach spaces. Acta Math., 130, 309–317.

• Semadeni, Z. (1982). Zbigniew Semadeni, Schauder bases in Banach spaces of continuous functions , (Vol. Springer 918). Lecture Notes in Mathematics. Mathematics Berlin: 918. Springer-Verlag. Berlin 2008.