Orthonormal basis

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In mathematics, an orthonormal basis of an inner product space V (i.e., a vector space with an inner product), is a set of basis vectors whose elements are mutually orthogonal and of magnitude 1 (unit vectors). 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.

For a finite-dimensional space, an orthonormal basis is a Hamel basis (a basis as defined in linear algebra, which spans the entire space).

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, not that its span equal the entire space.

An orthonormal basis of a vector space V makes no sense unless V is given an inner product; a Banach space does not have an orthonormal basis unless it is a Hilbert space.

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[edit] 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 {fn : n ∈ Z} with fn(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 {eb : b ∈ B} with eb(c) = 1 if b=c and 0 otherwise forms an orthonormal basis of l2(B).
  • Eigenfunctions of a Sturm-Liouville eigenproblem.

[edit] Basic formulae

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. 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.

[edit] 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.

[edit] 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.


[edit] See also