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 spaces L2([a, b]) or L2(Rn) 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 = ∑ <b, x> 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 and convenient description of its dual space (the space of all continuous linear functions from the space into the base field), which is itself a Hilbert 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
For a Hilbert space H, the continuous linear operators A : H -> H are of particular interest. Such a continuous operator is bounded in the sense that it maps bounded sets to bounded sets. This allows to define its norm as
The set L(H) of all continuous linear operators on H, together with the addition and composition operations, the norm and the adjoint operation, forms a C*-algebra; in fact, this is the motivating prototype and most important example of a C*-algebra.
An element A of L(H) is called self-adjoint or Hermitian if A* = A. These operators share many features of the real numbers and are sometimes seen as generalizations of them.
An element U of L(H) is called unitary if U is invertible and its inverse is given by U*. This can also be expressed by requiring that <Ux, Uy> = <x, y> for all x and y in H. The unitary operators form a group under composition, which can be viewed as the autormorphism group of H.
If S is a subset of the Hilbert space H, we define
In quantum mechanics, one also considers linear operators which need not be continuous and which need not be defined on the whole space H. One requires only that they are defined on a dense subspace of H. It is possible to define self-adjoint unbounded operators, and these play the role of the observables in the mathematical formulation of quantum mechanics.
Typical examples of self-adjoint unbounded operators on the Hilbert space L2(R) are given by the derivative Af = if ' (where i is the imaginary unit and f is a square integrable function) and by multiplication with x: Bf(x) = xf(x). These correspond to the momentum and position observables, respectively. Note that neither A nor B is defined on all of H, since in the case of A the derivative need not exist, and in the case of B the product function need not be square integrable. In both cases, the set of possible arguments form dense subspaces of L2(R).