Orthonormal Basis

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## Examples

## Basic formula

## Incomplete orthogonal sets

## Existence

## As a homogeneous space

## See also

## References

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Orthonormal Basis

In mathematics, particularly linear algebra, an **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.^{[1]}^{[2]}^{[3]} 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 Gram-Schmidt process.

In functional analysis, the concept of an orthonormal basis can be generalized to arbitrary (infinite-dimensional) inner product spaces.^{[4]} 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 [-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 *x ^{n}*.

- 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, and that each of their magnitudes equals one, ||*e*_{1}|| = ||*e*_{2}|| = ||*e*_{3}|| = 1. This means that is an orthonormal set. All vectors in**R**^{3}can be expressed as a sum of the basis vectors scaled- so 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 can be used to construct other orthogonal bases of .
- The set with forms an orthonormal basis of the space of functions with finite Lebesgue integrals, L
^{2}([0,1]), with respect to the 2-norm. This is fundamental to the study of Fourier series. - The set with if and 0 otherwise forms an orthonormal basis of l
^{2}(*B*). - Eigenfunctions of a Sturm-Liouville eigenproblem.
- An orthogonal matrix is a matrix whose column vectors form an orthonormal set.

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

When *B* is orthonormal, this simplifies to

and the square of 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*, and the formula is usually known as Parseval's identity.

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 such that

for all *x* and *y* in *H*.

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.

Using Zorn's lemma and the Gram-Schmidt process (or more simply well-ordering and transfinite recursion), one can show that *every* Hilbert space admits a basis, but not orthonormal base^{[5]}; 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).

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 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 for 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.

**^**Lay, David C. (2006).*Linear Algebra and Its Applications*(3rd ed.). Addison-Wesley. ISBN 0-321-28713-4.**^**Strang, Gilbert (2006).*Linear Algebra and Its Applications*(4th ed.). Brooks Cole. ISBN 0-03-010567-6.**^**Axler, Sheldon (2002).*Linear Algebra Done Right*(2nd ed.). Springer. ISBN 0-387-98258-2.**^**Rudin, Walter (1987).*Real & Complex Analysis*. McGraw-Hill. ISBN 0-07-054234-1.**^**Linear Functional Analysis Authors: Rynne, Bryan, Youngson, M.A. page 79

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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