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Hilbert space

In mathematics, a Hilbert space is a real or complex vector space with a positive-definite Hermitian form, that is complete under its norm. Thus it is an inner product space, which means that it has notions of distance and of angle (especially the notion of orthogonality or perpendicularity). The completeness requirement ensures that for infinite dimensional Hilbert spaces the limits exist when expected, which facilitates various definitions from calculus. A typical example of a Hilbert space is the space of square summable sequences.

Hilbert spaces allow simple geometric concepts, like projection and change of basis to be applied to infinite dimensional spaces, such as function spaces. They provide a context with which to formalize and generalize the concepts of the Fourier series in terms of arbitrary orthogonal polynomials and of the Fourier transform, which are central concepts from functional analysis. Hilbert spaces are of crucial importance in the mathematical formulation of quantum mechanics.

Older books and papers sometimes call a Hilbert space a unitary space or a linear space with an inner product, but this terminology is no longer used.

Introduction

Hilbert spaces were named after David Hilbert, who studied them in the context of integral equations. John von Neumann originated the designation " der abstrakte Hilbertsche Raum " in his famous work on unbounded Hermitian operators, published in 1929.^[1] Von Neumann was perhaps the mathematician who most clearly recognized their importance as a result of his seminal work on the foundations of quantum mechanics begun with Hilbert and Lothar (Wolfgang) Nordheim^[2] and continued with Eugene Wigner. The name "Hilbert space" was soon adopted by others, for example by Hermann Weyl in his 1931 book The Theory of Groups and Quantum Mechanics.^[3]

The elements of an abstract Hilbert space are sometimes called "vectors". All finite-dimensional inner product spaces (such as a real or complex vector space with the ordinary dot product) are Hilbert spaces. In applications, Hilbert spaces are typically sequences of complex numbers 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 for details. (The space of plane waves and bound states commonly used in quantum mechanics is known more formally as the rigged Hilbert space.)

Other applications include:

The theory of unitary group representations
The theory of square integrable stochastic processes
The Hilbert space theory of partial differential equations, in particular formulations of the Dirichlet problem
Spectral analysis of functions, including theories of wavelets

The inner product allows one to adopt a "geometrical" view and use geometrical language familiar from finite-dimensional spaces. Of all the infinite-dimensional topological vector spaces, the Hilbert spaces are the most "well-behaved" and the closest to the finite-dimensional spaces.

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. The Fourier transform then corresponds to a change of basis.

Definition

A Hilbert Space is an inner product space that is also a Banach space (a complete normed space) under the norm defined by the inner product.

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

$\|x\| = \sqrt{\langle x,x \rangle}.$

In any normed space, the open balls constitute a compatible topology; any normed vector space is a topological vector space (and even a uniform structure) and therefore so is any inner product space.

The Cauchy criterion may be defined for sequences in this space (as it can in any uniform space): a sequence {x_n} is a Cauchy sequence if for every positive real number ε there is a natural number N such that for all m, n > N, ||x_n – x_m|| < ε. H is a Hilbert space if it is complete with respect to this norm, that is if every Cauchy sequence converges to an element in the space. Thus, every Hilbert space is also a Banach space, but not vice versa.

Some authors use slightly different definitions. For example, Kolmogorov et al.^[4] define a Hilbert space as above but restrict the definition to separable and infinite-dimensional spaces. A separable, infinite-dimensional Hilbert space is unique up to isomorphism, called $\ell^2(\mathbb{N})\,$ [often written $\ell^2\,$ for shorthand — see the next section for the definition]. In this article, a Hilbert space is not assumed to be infinite-dimensional or separable.

Examples

In these examples, the underlying field of scalars is C, although the definitions apply to the case in which the underlying field of scalars is R.

Euclidean spaces

Cⁿ with the inner product definition

$\langle x, y \rangle = \sum_{k=1}^n x_k\overline{y_k}$

where the bar over a complex number denotes its complex conjugate.

Sequence spaces

Much more typical are the infinite-dimensional Hilbert spaces however. If B is any set, the sequence space l² (said "little ell two") over B is defined

$\ell^2(B) =\left\{ x:B \rightarrow \mathbb{C}\,\bigg|\,\sum_{b \in B} \left|x \left(b\right)\right|^2 < \infty \right\}$

This space becomes a Hilbert space with the inner product

$\langle x, y \rangle = \sum_{b \in B} x(b)\overline{y(b)}$

for all x and y in l²(B). B does not have to be a countable set in this definition, although if B is not countable, the resulting Hilbert space is not separable. In a sense made more precise below, every Hilbert space is isomorphic to one of the form l²(B) for a suitable set B. If B=N, the natural numbers, this space is simply called l².

Lebesgue spaces

These are function spaces associated to measure spaces (X, M, μ), where M is a σ-algebra of subsets of X and μ is a countably additive measure on M. Let L²_μ(X) be the space of complex-valued square-integrable measurable functions on X, modulo equality almost everywhere. Square integrable means the integral of the square of its absolute value is finite. Modulo equality almost everywhere means functions are identified if and only if they are equal outside of a set of measure 0.

The inner product of functions f and g is here given by

$\langle f,g\rangle=\int_X f(t) \overline{g(t)} \ d \mu(t)$

One needs to show:

That this integral indeed makes sense;
The resulting space is complete.

These facts are easy to derive; see, for example, Section 42 of Halmos (1950).^[5] Note that the use of the Lebesgue integral ensures that the space will be complete. See L^p space for further discussion of this example.

Sobolev spaces

Sobolev spaces, denoted by $H s$ or $W s,2$ , are another example of Hilbert spaces, and are used often in the field of partial differential equations.

Operations on Hilbert spaces

Two (or more) Hilbert spaces can be combined into a single Hilbert space by taking their direct sum or their tensor product.

Bases

An important concept is that of an orthonormal basis of a Hilbert space H: this is a family {e_k}_{k ∈ B} of H satisfying:

Elements are normalized: Every element of the family has norm 1: ||e_k|| = 1 for all k in B
Elements are orthogonal: Every two different elements of B are orthogonal: <e_k, e_j> = 0 for all k, j in B with k ≠ j.
Dense span: The linear span of B is dense in H.

An orthonormal basis is sometimes called an orthonormal sequence or orthonormal set.

Examples of orthonormal bases include:

the set {(1,0,0),(0,1,0),(0,0,1)} forms an orthonormal basis of R³
the sequence {f_n : n ∈ Z} with f_n(x) = exp(2πinx) forms an orthonormal basis of the complex space L²([0,1])
the family {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. That the span of the basis vectors is dense means that every vector in the space can be written as the limit of an infinite series and the orthogonality implies that this decomposition is unique.

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 infinite-dimensional separable Hilbert spaces are isomorphic, and since almost all Hilbert spaces used in physics are separable, when physicists talk about the Hilbert space they mean any separable one.

If {e_k}_{k ∈ B} is an orthonormal basis of H, then every element x of H may be written as

$x = \sum_{k \in B} \langle e_k , x \rangle e_k$

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 {e_k}_{k ∈ 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

$\langle \Phi \left(x\right), \Phi\left(y\right) \rangle = \langle x, y \rangle$

for all x and y in H.

Orthogonal complements and projections

If S is a subset of a Hilbert space H, the set of vectors orthogonal to S is defined by

$S^\mathrm{perp} = \left\{ x \in H : \langle x, s \rangle = 0\ \forall s \in S \right\}$

S^perp is a closed subspace of H and so forms itself a Hilbert space. If V is a closed subspace of H, then V^perp is called the orthogonal complement of V. In fact, every x in H can then be written uniquely as x = v + w, with v in V and w in V^perp. Therefore, H is the internal Hilbert direct sum of V and V^perp. The linear operator P_V : H → H which maps x to v is called the orthogonal projection onto V.

Theorem. The orthogonal projection P_V is a self-adjoint linear operator on H of norm ≤ 1 with the property P_V² = P_V. Moreover, any self-adjoint linear operator E such that E² = E is of the form P_V, where V is the range of E. For every x in H, P_V(x) is the unique element v of V which minimizes the distance ||x - v||.

This provides the geometrical interpretation of P_V(x): it is the best approximation to x by elements of V.

Reflexivity

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

$\phi \left(x\right) = \langle u, x \rangle$

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.

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

$\lVert A \rVert = \sup \left\{\,\lVert Ax \rVert : \lVert x \rVert \leq 1\,\right\}.$

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

$\langle A^* y, x \rangle = \langle y, Ax \rangle.$

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 automorphism group of H.

Unbounded operators

If a linear operator has a closed graph and is defined on all of a Hilbert space, then, by the closed graph theorem in Banach space theory, it is necessarily bounded. However, unbounded operators can be obtained by defining a linear map on a proper subspace of the Hilbert space.

In quantum mechanics, several interesting unbounded operators are defined on a dense subspace of Hilbert space. It is possible to define self-adjoint unbounded operators, and these play the role of the observables in the mathematical formulation of quantum mechanics.

Examples of self-adjoint unbounded operator on the Hilbert space L²(R) are:

A suitable extension of the differential operator

$[A f](x) = i \frac{d}{dx} f(x), \quad$

where i is the imaginary unit and f is a differentiable function of compact support.

The multiplication by x operator:

$[B f] (x) = xf(x).\quad$

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).

Notes and references

^ Von Neumann, John (1929). "Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren". Mathematische Annalen 102: 49–131.
^ Hilbert, David; Lothar Nordheim and John von Neumann (1927). "Über die Grundlagen der Quantenmechanik". Mathematische Annalen 98: 1–30.
^ Weyl, Hermann (1931). The Theory of Groups and Quantum Mechanics, English edition (1950), Dover Press. ISBN 0-486-60269-9.
^ Kolmogorov, Andrey; S. V. Fomin (1970). Introductory Real Analysis, Revised English edition, trans. by Richard A. Silverman (1975), Dover Press. ISBN 0-486-61226-0.
^ Halmos, Paul (1950). Measure Theory. D. van Nostrand Co.

Jean Dieudonné, Foundations of Modern Analysis, Academic Press, 1960.
B.M. Levitan, "Hilbert space" SpringerLink Encyclopaedia of Mathematics (2001)

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