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orthonormal basis

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**orthonormal basis**for an inner product space

*V*with finite dimension is a basis for

*V*whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other.BOOK, Lay, David C., Linear Algebra and Its Applications, Addisonâ€“Wesley, 2006, 3rd, 0-321-28713-4, BOOK, Strang, Gilbert, Gilbert Strang, Linear Algebra and Its Applications, Brooks Cole, 2006, 4th, 0-03-010567-6, BOOK, Axler, Sheldon, Linear Algebra Done Right, Springer Science+Business Media, Springer, 2002, 2nd, 0-387-98258-2, For example, the standard basis for a Euclidean space

**R**

*n**is an orthonormal basis, where the relevant inner product is the dot product of vectors. The image of the standard basis under a rotation or reflection (or any orthogonal transformation) is also orthonormal, and every orthonormal basis for**R**n*arises in this fashion.For a general inner product space

*V*, an orthonormal basis can be used to define normalized orthogonal coordinates on

*V*. Under these coordinates, the inner product becomes a dot product of vectors. Thus the presence of an orthonormal basis reduces the study of a finite-dimensional inner product space to the study of

**R**

*n*under dot product. Every finite-dimensional inner product space has an orthonormal basis, which may be obtained from an arbitrary basis using the Gramâ€“Schmidt process.In functional analysis, the concept of an orthonormal basis can be generalized to arbitrary (infinite-dimensional) inner product spaces.BOOK, Rudin, Walter, Walter Rudin, Real & Complex Analysis, McGraw-Hill, 1987, 0-07-054234-1, Given a pre-Hilbert space

*H*, an orthonormal basis for

*H*is an orthonormal set of vectors with the property that every vector in

*H*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

*H*. Note that an orthonormal basis in this sense is not generally a Hamel basis, since infinite linear combinations are required. Specifically, the linear span of the basis must be dense in

*H*, but it may not be the entire space.If we go on to Hilbert spaces, 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 [âˆ’1, 1] can be expressed (almost everywhere) as an infinite sum of Legendre polynomials (an orthonomal basis), but not necessarily as an infinite sum of the monomials

*xn*.

## Examples

- The set of vectors {
*e*1 = (1, 0, 0),*e*2 = (0, 1, 0),*e*3 = (0, 0, 1)} (the standard basis) forms an orthonormal basis of**R**3.

**Proof:**A straightforward computation shows that the inner products of these vectors equals zero, {{nowrap|1=

*e*1,

*e*2{{rangle}} =

*e*1,

*e*3{{rangle}} =

*e*2,

*e*3{{rangle}} = 0}} and that each of their magnitudes equals one, ||

*e*1|| = ||

*e*2|| = ||

*e*3|| = 1. This means that {{nowrap|{

*e*1,

*e*2,

*e*3} }} is an orthonormal set. All vectors {{nowrap|(

*x*,

*y*,

*z*)}} in

**R**3 can be expressed as a sum of the basis vectors scaled

(x,y,z) = xe_1 + ye_2 + ze_3, ,

so {{nowrap|{*e*1,

*e*2,

*e*3} }} spans

**R**3 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

**R**3.

- Notice that an orthogonal transformation of the standard inner-product space (mathbb{R}^n,langlecdot,cdotrangle) can be used to construct other orthogonal bases of mathbb{R}^n.
- The set {{nowrap|{
*f***'n****:**Z*n*âˆˆ**} }} with {{nowrap|1=***f**'n*(*x*) = exp(2Ï€*inx*)}} forms an orthonormal basis of the space of functions with finite Lebesgue integrals, L2([0,1]), with respect to the 2-norm. This is fundamental to the study of Fourier series. - The set {{nowrap|{
*e***'b****:***b*âˆˆ*B*} }} with {{nowrap|1=*e**'b*(*c*) = 1}} if {{nowrap|1=*b*=*c*}} and 0 otherwise forms an orthonormal basis of â„“2(*B*). - Eigenfunctions of a Sturmâ€“Liouville eigenproblem.
- An orthogonal matrix is a matrix whose column vectors form an orthonormal set.

## Basic formula

If*B*is an orthogonal basis of

*H*, then every element

*x*of

*H*may be written as

x=sum_{bin B}{langle x,brangleoverlVert brVert^2} b.

When *B*is orthonormal, this simplifies to

x=sum_{bin B}langle x,brangle b

and the square of the norm of *x*can be given by

|x|^2=sum_{bin B}|langle x,brangle |^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*, and the formula is usually known as Parseval's identity.If

*B*is an orthonormal basis of

*H*, then

*H*is

*isomorphic*to

*â„“*2(

*B*) in the following sense: there exists a bijective linear map {{nowrap|Î¦ :

*H*â†’

*â„“*2(

*B*)}} such that

langlePhi(x),Phi(y)rangle=langle x,yrangle

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 and the Gramâ€“Schmidt process (or more simply well-ordering and transfinite recursion), one can show that*every*Hilbert space admits a basis, but not orthonormal base Linear Functional Analysis Authors: Rynne, Bryan, Youngson, M.A. page 79; furthermore, any two orthonormal bases of the same space have the same cardinality (this can be proven in a manner akin to that of the proof of the usual dimension theorem for vector spaces, with separate cases depending on whether the larger basis candidate is countable or not). A Hilbert space is separable if and only if it admits a countable orthonormal basis. (One can prove this last statement without using the axiom of choice).

## As a homogeneous space

The set of orthonormal bases for a space is a principal homogeneous space for the orthogonal group O(*n*), and is called the Stiefel manifold V_n(mathbf{R}^n) of orthonormal

*n*-frames.In other words, the space of orthonormal bases is like the orthogonal group, but without a choice of base point: given an orthogonal space, there is no natural choice of orthonormal basis, but once one is given one, there is a one-to-one correspondence between bases and the orthogonal group.Concretely, a linear map is determined by where it sends a given basis: just as an invertible map can take any basis to any other basis, an orthogonal map can take any

*orthogonal*basis to any other

*orthogonal*basis.The other Stiefel manifolds V_k(mathbf{R}^n) for k < n of

*incomplete*orthonormal bases (orthonormal

*k*-frames) are still homogeneous spaces for the orthogonal group, but not

*principal*homogeneous spaces: any

*k*-frame can be taken to any other

*k*-frame by an orthogonal map, but this map is not uniquely determined.

## See also

## References

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