 Orthonormal basis

In mathematics, particularly linear algebra, an orthonormal basis for inner product space V with finite dimension is a basis for V whose vectors are orthonormal.^{[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 dot product of vectors. Thus the presence of an orthonormal basis reduces the study of a finitedimensional inner product space to the study of R^{n} under dot product. Every finitedimensional 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 (infinitedimensional) inner product spaces (or preHilbert spaces).^{[4]} Given a preHilbert 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.
Contents
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, <e_{1}, e_{2}> = <e_{1}, e_{3}> = <e_{2}, e_{3}> = 0 and that each of their magnitudes equals one, e_{1} = e_{2} = e_{3} = 1. This means {e_{1}, e_{2}, e_{3}} is an orthonormal set. All vectors (x, y, z) in R^{3} can be expressed as a sum of the basis vectors scaled
 so {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}.
 Proof: A straightforward computation shows that the inner products of these vectors equals zero, <e_{1}, e_{2}> = <e_{1}, e_{3}> = <e_{2}, e_{3}> = 0 and that each of their magnitudes equals one, e_{1} = e_{2} = e_{3} = 1. This means {e_{1}, e_{2}, e_{3}} is an orthonormal set. All vectors (x, y, z) in R^{3} can be expressed as a sum of the basis vectors scaled
 The set {f_{n} : n ∈ Z} with f_{n}(x) = exp(2πinx) 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.
 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
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 nonzero, and the expression is therefore welldefined. This sum is also called the Fourier expansion of x, and the formula is usually known as Parseval's identity. 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 and the Gram–Schmidt process (or more simply wellordering and transfinite recursion), 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 (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.
As a homogeneous space
Main article: Stiefel manifoldThe 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 nframes.
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 onetoone 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 k < n of incomplete orthonormal bases (orthonormal kframes) are still homogeneous spaces for the orthogonal group, but not principal homogeneous spaces: any kframe can be taken to any other kframe by an orthogonal map, but this map is not uniquely determined.
See also
 Basis (linear algebra)
 Schauder basis
 Gram–Schmidt
References
 ^ Lay, David C. (2006). Linear Algebra and Its Applications (3rd ed.). Addison–Wesley. ISBN 0321287134.
 ^ Strang, Gilbert (2006). Linear Algebra and Its Applications (4th ed.). Brooks Cole. ISBN 0030105676.
 ^ Axler, Sheldon (2002). Linear Algebra Done Right (2nd ed.). Springer. ISBN 0387982582.
 ^ Rudin, Walter (1987). Real & Complex Analysis. McGrawHill. ISBN 0070542341.
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