Orthonormal basis

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In mathematics, an orthonormal basis of an inner product space (i.e., a vector space with an inner product), or in particular of a Hilbert space, is a set of elements whose span is dense in the space, in which the elements are mutually orthogonal and normal, that is of length 1. Note that 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. This is why the definition given above required only that the span of an orthonormal basis be dense in the vector space, not that it equal the entire space. Nor is it possible to speak of an orthonormal basis of a vector space unless it first has an inner product; Banach spaces do not generally have orthonormal bases.

Examples of orthonormal bases include:

the set {(1,0,0),(0,1,0),(0,0,1)} forms an orthonormal basis of R³
the set {f_n : n ∈ Z} with f_n(x) = exp(2πinx) forms an orthonormal basis of the complex space L²([0,1])
the set {e_b : b ∈ B} with e_b(c) = 1 if b=c and 0 otherwise forms an orthonormal basis of l²(B).

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.)

Using Zorn's lemma, one can show that every Hilbert space admits 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.

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

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.

If B is an orthonormal basis of H, then H is isomorphic to l²(B) in the following sense: there exists a bijective linear map Φ : H -> l²(B) such that

for all x and y in H.