Hilbert space

A Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product (and is hence a Banach space). Hilbert spaces were named after David Hilbert, who studied them in the context of integral equations. Hilbert spaces serve to clarify and generalize the concept of Fourier expansion, certain linear transformations such as the Fourier transform, and are of crucial importance in the mathematical formulation of quantum mechanics. Hilbert spaces are studied in the branch of mathematics called functional analysis.

Examples

Examples of Hilbert spaces are Rⁿ and Cⁿ with the inner product definition

<math>\langle x, y \rangle = \sum x_k^* y_k</math>

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

<math>\langle f,g\rangle=\int f(t)^* g(t)\,dt</math>

The use of the Lebesgue integral ensures that the space will be complete. (One should bear in mind that by definition, a Lebesgue-integrable function is a Lebesgue-measurable function the integral of whose absolute value is finite. Thus, a function is not included in the Hilbert space L² unless the integral of the square of its absolute value is finite.)

If B is some set, we define l²(B) as the set of all functions x : B → R or C such that

<math>\sum_{b \in B} \left|x \left(b\right)\right|^2 < \infty</math>

This space becomes a Hilbert space if we define

<math>\langle x, y \rangle = \sum_{b \in B} x \left(b \right)^* y \left( b \right)</math>

for all x and y in l²(B). In a sense made more precise below, every Hilbert space is of the form l²(B) for a suitable set B.

Bases

An important concept is that of an orthonormal basis of a Hilbert space H: a subset B of H with three properties:

1. Every element of B has norm 1: <e, e> = 1 for all e in B
2. Every two different elements of B are orthogonal: <e, f> = 0 for all e, f in B with e ≠ f.
3. The linear span of B is dense in H.

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.

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

<math>x = \sum_{b \in B} \langle b, x \rangle b</math>

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

<math>\langle \Phi \left(x\right), \Phi\left(y\right) \rangle = \langle x, y \rangle</math>

for all x and y in H.

Reflexitivity

An important property of any Hilbert space is its reflexivity (see Banach space). 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

<math>\phi \left(x\right) = \langle u, x \rangle</math> for all x in H

and the association φ ↔ u provides an antilinear isomorphism between H and H'. This correspondence is exploited by the bra-ket notation popular in physics but frowned upon by mathematicians.

Bounded Operators

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

<math>\lVert A \rVert = \mathrm{sup}_{\lVert x \rVert \leq 1} \lVert Ax \rVert</math>

The sum and the composition of two continuous linear operators is again continuous and linear. For y in H, the map that sends x to <y, Ax> is linear and continuous, and according to the Riesz representation theorem can therefore be represented in the form

<math>\langle A^* y, x \rangle = \langle y, Ax \rangle</math>

This defines another continuous linear operator A^* : H → H, the adjoint of A.

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.

Orthogonal complements and projections

If S is a subset of the Hilbert space H, we define

<math>S^+ = \left\{ x \in H : \langle x, s \rangle = 0\ \forall s \in S \right\}</math>

The set S⁺ is a closed subspace of H and so forms itself a Hilbert space. If S is a closed subspace of H, then S⁺ is called the orthogonal complement of S because every x in H can then be written in a unique way as a sum

x = s + t

with s in S and t in S⁺. The function P : H → H which sends x to s is called the orthogonal projection on S. P is a self-adjoint continuous linear operator on H with the property P² = P, and any such operator is an orthogonal projection on some closed subspace. For every x in H, P(x) is that element of S which is closest to x.

Unbounded Operators

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).