Home > Orthonormal basis

These concepts are important both for finite-dimensional and infinite-dimensional spaces. For finite-dimensional spaces the condition of a dense span is the same as 'span', as used in linear algebra.

An orthonormal basis is not generally a "basis", i.e., it is not generally 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: the definition given above requires only that the span of an orthonormal basis be dense in the vector space, not that it equal the entire space.

An orthonormal basis of a vector space V makes no sense unless V is given an inner product; Banach spaces do not generally have orthonormal bases.

1 Examples

• The set {(1,0,0),(0,1,0),(0,0,1)} 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.

2 Basic formulae

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

Where B is orthonormal, we have instead

and the norm of x can be given by

.

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

for all x and y in H.

3 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.

4 Existence

Using Zorn's lemma, 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 cardinalityAlternative meaning: number of pitch classes in a set. In linguistics, cardinal numbers is the name given to number words that are used for quantity one two three , as opposed to ordinal numbers, words that are used for order first second third . See How. A Hilbert space is separable if and only if it admits a countable orthonormal basis.