Hilbert Space

All finite-dimensional inner product spaces (such as Euclidean space with the ordinary dot product) are Hilbert spaces. However, the infinite-dimensional examples are much more important in the applications, of which quantum mechanics is the most prominent one. The inner product allows to perform many "geometrical" construction familiar from finite dimensions also in the infinite-dimensional settings. Of all the infinite-dimensional topological vector spaces, the Hilbert spaces are the most "well-behaved" and the closest to the finite-dimensional spaces.

The elements of Hilbert spaces are sometimes called "vectors"; they are typically sequences or functions. In quantum mechanics for example, a physical system is described by a complex Hilbert space which contains the "wavefunctions" that stand for the possible states of the system. See mathematical formulation of quantum mechanics.

One goal of Fourier analysis is to write a given function as a (possibly infinite) sum of multiples of given base functions. This problem can be studied abstractly in Hilbert spaces: every Hilbert space has an orthonormal basis, and every element of the Hilbert space can be written in a unique way as a sum of multiples of these base elements.

Hilbert spaces were named after David Hilbert, who studied them in the context of integral equations. The definition however is due to John von Neumann.

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

A Hilbert space whose elements are sequences is given by l²: the elements are sequences (x_n) of real (or complex) numbers such that

• 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).

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.

Since all separable Hilbert spaces are isomorphic, and since almost all Hilbert spaces used in physics are separable, when physisists talk about the Hilbert space they mean any separable one.

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

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

Reflexitivity

An important property of any Hilbert space is its reflexivity. In fact, more is true: one has a complete and convenient description of its dual space (the space of all continuous linear functions from the space H 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 all x in H

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.

x = s + t