Hilbert space

In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. 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. They are studied in functional analysis.

Table of contents

1 Introduction
2 Examples
3 Bases
4 Reflexitivity
5 Bounded Operators
6 Orthogonal complements and projections
7 Unbounded Operators

Introduction

Every inner product <.,.> on a real or complex vector space H gives rise to a norm ||.|| as follows:

We call H a Hilbert space if it is complete with respect to this norm. Completeness in this context means that any Cauchy sequence of elements of the space converges to an element in the space, in the sense that the norm of differences approaches zero. Every Hilbert space is thus also a Banach space (but not vice versa).

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.

Examples

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

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

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.) See L^p space for further discussion of this example.

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 inner product of x = (x_n) and y = (y_n) is defined by

More generally, if B is any set, we define l²(B) as the set of all functions x : B → R or C such that

This space becomes a Hilbert space if we define

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:

Every element of B has norm 1: <e, e> = 1 for all e in B
Every two different elements of B are orthogonal: <e, f> = 0 for all e, f in B with e ≠ f.
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 basis is also called a Hamel basis.

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

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.

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

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

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

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

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).

Need to mention Spectrum of an operator, spectral theorem