Hilbert spaces

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 and certain linear transformations such as the Fourier transform. Hilbert spaces are of crucial importance in the mathematical formulation of quantum mechanics, although many basic features of quantum mechanics can be understood without going into details about Hilbert spaces. Hilbert spaces are studied in functional analysis.

Introduction

Hilbert spaces were named after David Hilbert, who studied them in the context of integral equations. The origin of the designation "der abstrakte Hilbertsche Raum" is John von Neumann in his famous work on unbounded Hermitian operators published in 1929. 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 and continued with Eugene Wigner. The name "Hilbert space" was soon adopted by others, for example by Hermann Weyl in his book The Theory of Groups and Quantum Mechanics published in 1931 (English language paperback ISBN 0486602699).

The elements of an abstract Hilbert space are sometimes called "vectors". In applications, they 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 Hilbert space of plane waves and bound states commonly used in quantum mechanics is known more formally as the rigged Hilbert space.

Definition

Every inner product ⟨., .⟩ on a real or complex vector space H gives rise to a norm ∣∣.∣∣as follows:

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

We call H a Hilbert space if it is complete with respect to this norm. Completeness in this context means that every 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 applications. These 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
• Mathematical formulations of quantum mechanics

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.

Examples

In these examples, we will assume the underlying field of scalars is C, although the definitions apply to the case the underlying field of scalars is R.

Euclidean spaces

Cn with the inner product definition

$$\langle x, y \rangle = \sum_{k=1}^n \overline{x_k} y_k$$ 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, we define the sequence space little l2 over B, denoted by

$$\ell^2(B) =\left\{ x:B \rightarrow \mathbb{C}\,\bigg|\,\sum_{b \in B} \left|x \left(b\right)\right|^2 < \infty \right\}$$ This space becomes a Hilbert space with the inner product

$$\langle x, y \rangle = \sum_{b \in B} \overline{x(b)} y(b)$$ 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 the form l2(B) for a suitable set B. If B=N, we write simply 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 L2μ(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

$$\langle f,g\rangle=\int_X \overline{f(t)} g(t) \ d \mu(t)$$ One needs to show:

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

These are technically easy facts, and the interested reader should consult the Halmos reference below, Section 42. Note that the use of the Lebesgue integral ensures that the space will be complete. See Lp space for further discussion of this example.

Sobolev spaces

Sobolev spaces, denoted by Hs or Ws, 2, are another example of Hilbert spaces, and are used very often in the field of Partial differential equations.

Operations on Hilbert spaces

Given two (or more) Hilbert spaces, we can combine them into a big 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 {ek}kB of H satisfying:

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

We also use the expressions orthonormal sequence and 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 {fn : nZ} with fn(x) = exp(2πinx) forms an orthonormal basis of the complex space L2([0,1])
• the family {eb : bB} with eb(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 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 {ek}kB is an orthonormal basis of H, then every element x of H may be written as

x = ∑k ∈ Bek, xek

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 {ek}kB is an orthonormal basis of H, then H is isomorphic to l2(B) in the following sense: there exists a bijective linear map Φ : Hl2(B) such that

⟨Φ(x), Φ(y)⟩ = ⟨x, y⟩ for all x and y in H.

Orthogonal complements and projections

If S is a subset of a Hilbert space H, we define the set of vectors orthogonal to S

Sperp = {x ∈ H : ⟨x, s⟩ = 0 ∀s ∈ S}

Sperp is a closed subspace of H and so forms itself a Hilbert space. If V is a closed subspace of H, then Vperp 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 Vperp. Therefore, H is the internal Hilbert direct sum of V and Vperp. The linear operator PV : HH which maps x to v is called the orthogonal projection onto V.

Theorem. The orthogonal projection PV is a self-adjoint linear operator on H of norm ≤ 1 with the property PV2 = PV. Moreover, any self-adjoint linear operator E such that E2 = E is of the form PV, where V is the range of E. For every x in H, PV(x) is the unique element v of V which minimizes the distance ||x - v||.

This provides the geometrical interpretation of PV(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

ϕ(x) = ⟨u, x⟩ 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 : HH 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

A∥ = sup{ ∥Ax∥ : ∥x∥ ≤ 1 }.

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

A * y, x⟩ = ⟨y, Ax⟩.

This defines another continuous linear operator A* : HH, 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 is defined on all of a Hilbert space then it is necessarily bounded. However, if we allow ourselves to define a linear map that is defined on a proper subspace of the Hilbert space, then we can obtain unbounded operators.

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:
[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).

References

• Jean Dieudonné, Foundations of Modern Analysis, Academic Press, 1960.
• Paul Halmos, Measure Theory, D. van Nostrand Co, 1950.
• David Hilbert, Lothar Nordheim, and John von Neumann, "Űber die Grundlagen der Quantenmechanik," Mathematische Annalen, volume 98, pages 1-30, 1927.
• John von Neumann, "Allgemiene Eigenwerttheorie Hermitescher Funktionaloperatoren," Mathematische Annalen, volume 102, pages 49-131, 1929.
• Hermann Weyl, The Theory of Groups and Quantum Mechanics, Dover Press, 1950. This book was originally published in German in 1931.

Category:Linear Algebra