Orthonormal basis

• For ${\displaystyle \mathbb {R} ^{3}}$ , the set of vectors ${\displaystyle \left\{\mathbf {e_{1}} ={\begin{pmatrix}1&0&0\end{pmatrix}}\ ,\ \mathbf {e_{2}} ={\begin{pmatrix}0&1&0\end{pmatrix}}\ ,\ \mathbf {e_{3}} ={\begin{pmatrix}0&0&1\end{pmatrix}}\right\},}$  is called the standard basis and forms an orthonormal basis of ${\displaystyle \mathbb {R} ^{3}}$  with respect to the standard dot product. Note that both the standard basis and standard dot product rely on viewing ${\displaystyle \mathbb {R} ^{3}}$  as the Cartesian product ${\displaystyle \mathbb {R} \times \mathbb {R} \times \mathbb {R} }$ 

  Proof: A straightforward computation shows that the inner products of these vectors equals zero, ${\displaystyle \left\langle \mathbf {e_{1}} ,\mathbf {e_{2}} \right\rangle =\left\langle \mathbf {e_{1}} ,\mathbf {e_{3}} \right\rangle =\left\langle \mathbf {e_{2}} ,\mathbf {e_{3}} \right\rangle =0}$  and that each of their magnitudes equals one, ${\displaystyle \left\|\mathbf {e_{1}} \right\|=\left\|\mathbf {e_{2}} \right\|=\left\|\mathbf {e_{3}} \right\|=1.}$  This means that ${\displaystyle \left\{\mathbf {e_{1}} ,\mathbf {e_{2}} ,\mathbf {e_{3}} \right\}}$  is an orthonormal set. All vectors ${\displaystyle (\mathbf {x} ,\mathbf {y} ,\mathbf {z} )\in \mathbb {R} ^{3}}$  can be expressed as a sum of the basis vectors scaled $${\displaystyle (\mathbf {x} ,\mathbf {y} ,\mathbf {z} )=\mathbf {xe_{1}} +\mathbf {ye_{2}} +\mathbf {ze_{3}} ,}$$  so ${\displaystyle \left\{\mathbf {e_{1}} ,\mathbf {e_{2}} ,\mathbf {e_{3}} \right\}}$  spans ${\displaystyle \mathbb {R} ^{3}}$  and hence must be a basis. It may also be shown that the standard basis rotated about an axis through the origin or reflected in a plane through the origin also forms an orthonormal basis of ${\displaystyle \mathbb {R} ^{3}}$ .

• For ${\displaystyle \mathbb {R} ^{n}}$ , the standard basis and inner product are similarly defined. Any other orthonormal basis is related to the standard basis by an orthogonal transformation in the group O(n).
• For pseudo-Euclidean space ${\displaystyle \mathbb {R} ^{p,q},}$ , an orthogonal basis ${\displaystyle \{e_{\mu }\}}$  with metric ${\displaystyle \eta }$  instead satisfies ${\displaystyle \eta (e_{\mu },e_{\nu })=0}$  if ${\displaystyle \mu \neq \nu }$ , ${\displaystyle \eta (e_{\mu },e_{\mu })=+1}$  if ${\displaystyle 1\leq \mu \leq p}$ , and ${\displaystyle \eta (e_{\mu },e_{\mu })=-1}$  if ${\displaystyle p+1\leq \mu \leq p+q}$ . Any two orthonormal bases are related by a pseudo-orthogonal transformation. In the case ${\displaystyle (p,q)=(1,3)}$ , these are Lorentz transformations.
• The set ${\displaystyle \left\{f_{n}:n\in \mathbb {Z} \right\}}$  with ${\displaystyle f_{n}(x)=\exp(2\pi inx),}$  where ${\displaystyle \exp }$  denotes the exponential function, forms an orthonormal basis of the space of functions with finite Lebesgue integrals, ${\displaystyle L^{2}([0,1]),}$  with respect to the 2-norm. This is fundamental to the study of Fourier series.
• The set ${\displaystyle \left\{e_{b}:b\in B\right\}}$  with ${\displaystyle e_{b}(c)=1}$  if ${\displaystyle b=c}$  and ${\displaystyle e_{b}(c)=0}$  otherwise forms an orthonormal basis of ${\displaystyle \ell ^{2}(B).}$ 
• Eigenfunctions of a Sturm–Liouville eigenproblem.
• The column vectors of an orthogonal matrix form an orthonormal set.

If ${\displaystyle B}$  is an orthogonal basis of ${\displaystyle H,}$  then every element ${\displaystyle x\in H}$  may be written as $${\displaystyle x=\sum _{b\in B}{\frac {\langle x,b\rangle }{\lVert b\rVert ^{2}}}b.}$$ 

When ${\displaystyle B}$  is orthonormal, this simplifies to $${\displaystyle x=\sum _{b\in B}\langle x,b\rangle b}$$  and the square of the norm of ${\displaystyle x}$  can be given by $${\displaystyle \|x\|^{2}=\sum _{b\in B}|\langle x,b\rangle |^{2}.}$$ 

Even if ${\displaystyle B}$  is uncountable, only countably many terms in this sum will be non-zero, and the expression is therefore well-defined. This sum is also called the Fourier expansion of ${\displaystyle x,}$  and the formula is usually known as Parseval's identity.

If ${\displaystyle B}$  is an orthonormal basis of ${\displaystyle H,}$  then ${\displaystyle H}$  is isomorphic to ${\displaystyle \ell ^{2}(B)}$  in the following sense: there exists a bijective linear map ${\displaystyle \Phi :H\to \ell ^{2}(B)}$  such that $${\displaystyle \langle \Phi (x),\Phi (y)\rangle =\langle x,y\rangle \ \ \forall \ x,y\in H.}$$ 

A set ${\displaystyle S}$  of mutually orthonormal vectors in a Hilbert space ${\displaystyle H}$  is called an orthonormal system. An orthonormal basis is an orthonormal system with the additional property that the linear span of ${\displaystyle S}$  is dense in ${\displaystyle H}$ .^[6] Alternatively, the set ${\displaystyle S}$  can be regarded as either complete or incomplete with respect to ${\displaystyle H}$ . That is, we can take the smallest closed linear subspace ${\displaystyle V\subseteq H}$  containing ${\displaystyle S.}$  Then ${\displaystyle S}$  will be an orthonormal basis of ${\displaystyle V;}$  which may of course be smaller than ${\displaystyle H}$  itself, being an incomplete orthonormal set, or be ${\displaystyle H,}$  when it is a complete orthonormal set.

Using Zorn's lemma and the Gram–Schmidt process (or more simply well-ordering and transfinite recursion), one can show that every Hilbert space admits an orthonormal basis;^[7] furthermore, any two orthonormal bases of the same space have the same cardinality (this can be proven in a manner akin to that of the proof of the usual dimension theorem for vector spaces, with separate cases depending on whether the larger basis candidate is countable or not). A Hilbert space is separable if and only if it admits a countable orthonormal basis. (One can prove this last statement without using the axiom of choice. However, one would have to use the axiom of countable choice.)

For concreteness we discuss orthonormal bases for a real, ${\displaystyle n}$ -dimensional vector space ${\displaystyle V}$  with a positive definite symmetric bilinear form ${\displaystyle \phi =\langle \cdot ,\cdot \rangle }$ .

One way to view an orthonormal basis with respect to ${\displaystyle \phi }$  is as a set of vectors ${\displaystyle {\mathcal {B}}=\{e_{i}\}}$ , which allow us to write ${\displaystyle v=v^{i}e_{i}\ \ \forall \ v\in V}$  , and ${\displaystyle v^{i}\in \mathbb {R} }$  or ${\displaystyle (v^{i})\in \mathbb {R} ^{n}}$ . With respect to this basis, the components of ${\displaystyle \phi }$  are particularly simple: ${\displaystyle \phi (e_{i},e_{j})=\delta _{ij}}$  (where ${\displaystyle \delta _{ij}}$  is the Kronecker delta).

We can now view the basis as a map ${\displaystyle \psi _{\mathcal {B}}:V\rightarrow \mathbb {R} ^{n}}$  which is an isomorphism of inner product spaces: to make this more explicit we can write

${\displaystyle \psi _{\mathcal {B}}:(V,\phi )\rightarrow (\mathbb {R} ^{n},\delta _{ij}).}$ 

Explicitly we can write ${\displaystyle (\psi _{\mathcal {B}}(v))^{i}=e^{i}(v)=\phi (e_{i},v)}$  where ${\displaystyle e^{i}}$  is the dual basis element to ${\displaystyle e_{i}}$ .

The inverse is a component map

${\displaystyle C_{\mathcal {B}}:\mathbb {R} ^{n}\rightarrow V,(v^{i})\mapsto \sum _{i=1}^{n}v^{i}e_{i}.}$ 

These definitions make it manifest that there is a bijection

${\displaystyle \{{\text{Space of orthogonal bases }}{\mathcal {B}}\}\leftrightarrow \{{\text{Space of isomorphisms }}V\leftrightarrow \mathbb {R} ^{n}\}.}$ 

The space of isomorphisms admits actions of orthogonal groups at either the ${\displaystyle V}$  side or the ${\displaystyle \mathbb {R} ^{n}}$  side. For concreteness we fix the isomorphisms to point in the direction ${\displaystyle \mathbb {R} ^{n}\rightarrow V}$ , and consider the space of such maps, ${\displaystyle {\text{Iso}}(\mathbb {R} ^{n}\rightarrow V)}$ .

This space admits a left action by the group of isometries of ${\displaystyle V}$ , that is, ${\displaystyle R\in {\text{GL}}(V)}$  such that ${\displaystyle \phi (\cdot ,\cdot )=\phi (R\cdot ,R\cdot )}$ , with the action given by composition: ${\displaystyle R*C=R\circ C.}$ 

This space also admits a right action by the group of isometries of ${\displaystyle \mathbb {R} ^{n}}$ , that is, ${\displaystyle R_{ij}\in {\text{O}}(n)\subset {\text{Mat}}_{n\times n}(\mathbb {R} )}$ , with the action again given by composition: ${\displaystyle C*R_{ij}=C\circ R_{ij}}$ .

The set of orthonormal bases for ${\displaystyle \mathbb {R} ^{n}}$  with the standard inner product is a principal homogeneous space or G-torsor for the orthogonal group ${\displaystyle G={\text{O}}(n),}$  and is called the Stiefel manifold ${\displaystyle V_{n}(\mathbb {R} ^{n})}$  of orthonormal ${\displaystyle n}$ -frames.^[8]

In other words, the space of orthonormal bases is like the orthogonal group, but without a choice of base point: given the space of orthonormal bases, there is no natural choice of orthonormal basis, but once one is given one, there is a one-to-one correspondence between bases and the orthogonal group. Concretely, a linear map is determined by where it sends a given basis: just as an invertible map can take any basis to any other basis, an orthogonal map can take any orthogonal basis to any other orthogonal basis.

The other Stiefel manifolds ${\displaystyle V_{k}(\mathbb {R} ^{n})}$  for ${\displaystyle k<n}$  of incomplete orthonormal bases (orthonormal ${\displaystyle k}$ -frames) are still homogeneous spaces for the orthogonal group, but not principal homogeneous spaces: any ${\displaystyle k}$ -frame can be taken to any other ${\displaystyle k}$ -frame by an orthogonal map, but this map is not uniquely determined.

• The set of orthonormal bases for ${\displaystyle \mathbb {R} ^{p,q}}$  is a G-torsor for ${\displaystyle G={\text{O}}(p,q)}$ .
• The set of orthonormal bases for ${\displaystyle \mathbb {C} ^{n}}$  is a G-torsor for ${\displaystyle G={\text{U}}(n)}$ .
• The set of orthonormal bases for ${\displaystyle \mathbb {C} ^{p,q}}$  is a G-torsor for ${\displaystyle G={\text{U}}(p,q)}$ .
• The set of right-handed orthonormal bases for ${\displaystyle \mathbb {R} ^{n}}$  is a G-torsor for ${\displaystyle G={\text{SO}}(n)}$ 

• Orthogonal basis
• Basis (linear algebra) – Set of vectors used to define coordinates
• Orthonormal frame – Euclidean space without distance and angles
• Schauder basis – Computational tool
• Total set – subset of a topological vector space whose linear span is densePages displaying wikidata descriptions as a fallback

• Roman, Stephen (2008). Advanced Linear Algebra. Graduate Texts in Mathematics (Third ed.). Springer. ISBN 978-0-387-72828-5. (page 218, ch.9)
• Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
• Steinwart, Ingo; Christmann, Andreas (2008). Support vector machines. New York: Springer. doi:10.1007/978-0-387-77242-4. ISBN 978-0-387-77241-7.

• This Stack Exchange Post discusses why the set of Dirac Delta functions is not a basis of L²([0,1]).