Orthonormal basis

In mathematics, an orthonormal basis of an inner product space V (i.e., a vector space with an inner product), is a set of mutually orthogonal vectors of magnitude 1 (unit vectors) that span the space when infinite linear combinations are allowed. (In some contexts, especially in linear algebra, the concept of basis (linear algebra) means a set of vectors that span a space when only finite linear combinations are allowed.) Such an infinite linear combination is an infinite series, and the concept of convergence relied upon is defined in terms of the space's inner product.

Elements in an orthogonal basis do not have to be unit vectors, but must be mutually perpendicular. It is easy to change the vectors in an orthogonal basis by scalar multiples to get an orthonormal basis, and indeed this is a typical way that an orthonormal basis is constructed.

The standard basis of the n-dimensional Euclidean space Rⁿ is an example of orthonormal (and ordered) basis.

For a finite-dimensional space, every orthonormal basis is a Hamel basis (a basis as defined in linear algebra, that spans the entire space), but most Hamel bases are not orthonormal bases. For an (infinite-dimensional) Hilbert space, an orthonormal basis is not a Hamel basis, i.e., it is not possible to write every member of the space as a linear combination of finitely many members of an orthonormal basis. In the infinite-dimensional case the distinction matters. An orthonormal basis of a Hilbert space H is required to have a dense linear span in H, but its linear span is not the entire space.

An orthonormal basis of a vector space V makes no sense unless V is given an inner product. In a Banach space that is not an inner-product space it makes no sense to speak of whether a set of vectors is orthonormal.

Examples

The set {e₁ = (1, 0, 0), e₂ = (0, 1, 0), e₃ = (0, 0, 1)} (the standard basis) forms an orthonormal basis of R³.

Proof: A straightforward computation shows that <e₁, e₂> = <e₁, e₃> = <e₂, e₃> = 0 and that ||e₁|| = ||e₂|| = ||e₃|| = 1. So {e₁, e₂, e₃} is an orthonormal set. For all (x, y, z) in R³ we have

(x,y,z)=xe_{1}+ye_{2}+ze_{3},\,

so {e₁,e₂,e₃} spans R³ 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 forms an orthonormal basis of R³.

The set {f_n : n ∈ Z} with f_n(x) = exp(2πinx) forms an orthonormal basis of the complex space L²([0,1]). This is fundamental to the study of Fourier series.
The set {e_b : b ∈ B} with e_b(c) = 1 if b = c and 0 otherwise forms an orthonormal basis of ℓ²(B).
Eigenfunctions of a Sturm–Liouville eigenproblem.

Basic formula

If B is an orthogonal basis of H, then every element x of H may be written as

x=\sum _{b\in B}{\langle x,b\rangle  \over \lVert b\rVert ^{2}}b.

When B is orthonormal, we have instead

x=\sum _{b\in B}\langle x,b\rangle b

and the norm of x can be given by

\|x\|^{2}=\sum _{b\subset B}|\langle x,b\rangle |^{2}.

Even if 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 x, and the formula is usually known as Parseval's identity. See also Generalized Fourier series.

If B is an orthonormal basis of H, then H is isomorphic to ℓ²(B) in the following sense: there exists a bijective linear map Φ : H -> ℓ²(B) such that

\langle \Phi (x),\Phi (y)\rangle =\langle x,y\rangle

for all x and y in H.

Incomplete orthogonal sets

Given a Hilbert space H and a set S of mutually orthogonal vectors in H, we can take the smallest closed linear subspace V of H containing S. Then S will be an orthogonal basis of V; which may of course be smaller than H itself, being an incomplete orthogonal set, or be H, when it is a complete orthogonal set.

Existence

Using Zorn's lemma and the Gram–Schmidt process, one can show that every Hilbert space admits a basis and thus an orthonormal basis; furthermore, any two orthonormal bases of the same space have the same cardinality. A Hilbert space is separable if and only if it admits a countable orthonormal basis.

As a homogeneous space

The set of orthonormal bases for a space is a principal homogeneous space for the orthogonal group O(n), and is called the Stiefel manifold $V_{n}(\mathbf {R} ^{n})$ of orthonormal n-frames.

In other words, the space of orthonormal bases is like the orthogonal group, but without a choice of base point: given an orthogonal space, 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 $V_{k}(\mathbf {R} ^{n})$ for $k<n$ of incomplete orthonormal bases (orthonormal k-frames) are still homogeneous spaces for the orthogonal group, but not principal homogeneous spaces: any k-frame can be taken to any other k-frame by an orthogonal map, but this map is not uniquely determined.