Examples of Hilbert spaces are Rn and Cn with the inner product definition <x, y> = ∑ xk yk*, where * denotes complex conjugation. Much more typical are the infinite dimensional Hilbert spaces however, in particular the space L2([a, b]) of square Lebesgue-integrable functions with values in R or C, modulo the subspace of those functions whose square integral is zero. The inner product of the two functions f and g is here given by
An important concept is that of an orthonormal basis of a Hilbert space H: a subset B of H with three properties:
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.
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
x = ∑ <x,b> b b∈B
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.
An important fact is that every Hilbert space is reflexive (see Banach space) and that one has a complete description of its dual space. Indeed, the [Riesz representation theorem]? states that to every element φ of the dual H' there exists one and only one u in H such that