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 |

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:

*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 **R**^{n} and **C**^{n} with the inner product definition

^{*}denotes complex conjugation.

Much more typical are the infinite dimensional Hilbert spaces however, in particular the spaces L^{2}([*a*, *b*]) or L^{2}(**R**^{n}) 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

^{2}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 *l*^{2}: the elements are sequences (*x*_{n}) of real (or complex) numbers such that

*x*= (

*x*

_{n}) and

*y*= (

*y*

_{n}) is defined by

*B*is any set, we define

*l*

^{2}(

*B*) as the set of all functions

*x*:

*B*→

**R**or

**C**such that

*x*and

*y*in

*l*

^{2}(

*B*). In a sense made more precise below, every Hilbert space is of the form

*l*

^{2}(

*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*.

- the set {(1,0,0),(0,1,0),(0,0,1)} forms an orthonormal basis of
**R**^{3} - the set {
*f*_{n}:*n*∈**Z**} with*f*_{n}(*x*) = exp(2π*inx*) forms an orthonormal basis of the complex space L^{2}([0,1]) - the set {
*e*_{b}:*b*∈*B*} with*e*_{b}(*c*) = 1 if*b*=*c*and 0 otherwise forms an orthonormal basis of*l*^{2}(*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

*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 *l*^{2}(*B*) in the following sense: there exists a bijective linear map Φ : *H* → *l*^{2}(*B*) such that

*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*

*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

*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

*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

*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*

*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

*P*=

^{2}*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 L^{2}(**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 L^{2}(**R**).

*Need to mention Spectrum of an operator, spectral theorem*

See also mathematical analysis, functional analysis, harmonic analysis.