# Orthonormal basis

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

*basis, and indeed this is a typical way that an orthonormal basis is constructed, via an orthogonal basis.*

**orthonormal**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.

## Contents

## Examples

- The set {(1,0,0),(0,1,0),(0,0,1)} (the standard basis), as well as versions obtained by rotation about an axis through the origin or reflection in a plane through the origin, or a combination, each form an orthonormal basis of
**R**^{3} - The set {
*f*_{n}:*n*∈**Z**} with*f*_{n}(*x*) = exp(2π*inx*) forms an orthonormal basis of the complex space L^{2}([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*l*^{2}(*B*). - Eigenfunctions of a Sturm-Liouville eigenproblem.

## 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 *l*^{2}(*B*) in the following sense: there exists a bijective linear map Φ : *H* `->` *l*^{2}(*B*) such that

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, 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.

## 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).

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