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), or in particular of a Hilbert space H, is a set of elements whose span is dense in the space, in which the elements are mutually orthogonal and normal, that is, of magnitude 1. An orthogonal basis satisfies the same conditions, without the condition of length 1; 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, via an orthogonal 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; a Banach space does not have an orthonormal basis unless it is a Hilbert space.

Contents 1 Examples 2 Basic formulae 3 Incomplete orthogonal sets 4 Existence 5 Relation to Hamel bases

[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

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.

[edit] Basic formulae

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

When 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.

[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] Relation to Hamel bases

Note that in the infinite-dimensional case, an orthonormal basis will not be a basis in the sense of linear algebra; to distinguish the two, the latter bases are also called Hamel bases. (Hamel bases are of little practical interest in inner product spaces, while orthonormal bases are of major importance - the distinction may though shed light on what an orthonormal basis is.)

See also Basis (linear algebra) and Schauder basis.