Editing Orthonormal basis (section)

{{Short description|Specific linear basis (mathematics)}}

In [[mathematics]], particularly [[linear algebra]], an '''orthonormal basis''' for an [[inner product space]] <math>V</math> with finite [[Dimension (linear algebra)|dimension]] is a [[Basis (linear algebra)|basis]] for <math>V</math> whose vectors are [[orthonormal]], that is, they are all [[unit vector]]s and [[Orthogonality_(mathematics)|orthogonal]] to each other.<ref>{{cite book|last=Lay|first=David C.|title=Linear Algebra and Its Applications|url=https://archive.org/details/studyguidetoline0000layd|url-access=registration|publisher=[[Addison–Wesley]]|year=2006|edition = 3rd|isbn=0-321-28713-4}}</ref><ref>{{cite book|last=Strang|first=Gilbert|author-link=Gilbert Strang|title=Linear Algebra and Its Applications|publisher=[[Brooks Cole]]|year=2006|edition = 4th|isbn=0-03-010567-6}}</ref><ref>{{cite book|last = Axler|first = Sheldon|title = Linear Algebra Done Right|publisher = [[Springer Science+Business Media|Springer]]|year = 2002|edition = 2nd|isbn = 0-387-98258-2}}</ref> For example, the [[standard basis]] for a [[Euclidean space]] <math>\R^n</math> is an orthonormal basis, where the relevant inner product is the [[dot product]] of vectors. The [[Image (mathematics)|image]] of the standard basis under a [[Rotation (mathematics)|rotation]] or [[Reflection (mathematics)|reflection]] (or any [[orthogonal transformation]]) is also orthonormal, and every orthonormal basis for <math>\R^n</math> arises in this fashion.
An orthonormal basis can be derived from an [[orthogonal basis]] via [[Normalize (linear algebra)|normalization]].
{{anchor|Frame}}The choice of an [[origin (mathematics)|origin]] and an orthonormal basis forms a [[coordinate frame]] known as an '''''orthonormal frame'''''.

For a general inner product space <math>V,</math> an orthonormal basis can be used to define normalized [[orthogonal coordinates]] on <math>V.</math> Under these coordinates, the inner product becomes a dot product of vectors. Thus the presence of an orthonormal basis reduces the study of a [[Dimension (vector space)|finite-dimensional]] inner product space to the study of <math>\R^n</math> under the dot product. Every finite-dimensional inner product space has an orthonormal basis, which may be obtained from an arbitrary basis using the [[Gram–Schmidt process]].

In [[functional analysis]], the concept of an orthonormal basis can be generalized to arbitrary (infinite-dimensional) [[inner product space]]s.<ref>{{cite book|last=Rudin|first=Walter|author-link=Walter Rudin|title=Real & Complex Analysis|publisher=[[McGraw-Hill]]|year=1987|isbn=0-07-054234-1}}</ref> Given a pre-Hilbert space <math>H,</math> an ''orthonormal basis'' for <math>H</math> is an orthonormal set of vectors with the property that every vector in <math>H</math> can be written as an [[infinite linear combination]] of the vectors in the basis. In this case, the orthonormal basis is sometimes called a '''Hilbert basis''' for <math>H.</math> Note that an orthonormal basis in this sense is not generally a [[Hamel basis]], since infinite linear combinations are required.{{sfn|Roman|2008|p=218|loc=ch. 9}} Specifically, the [[linear span]] of the basis must be [[Dense set|dense]] in <math>H,</math> although not necessarily the entire space.

If we go on to [[Hilbert space]]s, a non-orthonormal set of vectors having the same linear span as an orthonormal basis may not be a basis at all. For instance, any [[square-integrable function]] on the interval <math>[-1,1]</math> can be expressed ([[almost everywhere]]) as an infinite sum of [[Legendre polynomials]] (an orthonormal basis), but not necessarily as an infinite sum of the [[monomial]]s <math>x^n.</math>

A different generalisation is to pseudo-inner product spaces, finite-dimensional vector spaces <math>M</math> equipped with a non-degenerate [[symmetric bilinear form]] known as the [[metric tensor]]. In such a basis, the metric takes the form <math>\text{diag}(+1,\cdots,+1,-1,\cdots,-1)</math> with <math>p</math> positive ones and <math>q</math> negative ones.