Hilbert space

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For the Hilbert space-filling curve, see Hilbert curve.

This article assumes some familiarity with analytic geometry and the concept of a limit. The article on vector spaces contains useful background, and the article on functional analysis is closely related.

The mathematical concept of a Hilbert space, named after the German mathematician David Hilbert, generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the two-dimensional plane and three-dimensional space to infinite-dimensional spaces. In more formal terms, a Hilbert space is an inner product space — an abstract vector space in which distances and angles can be measured — which is "complete", meaning that if a sequence of vectors approaches a limit, then that limit is guaranteed to be in the space as well.

Hilbert spaces arise naturally and frequently in mathematics, physics, and engineering, typically as infinite-dimensional function spaces. They are indispensable tools in the theories of partial differential equations, quantum mechanics, and signal processing. The recognition of a common algebraic structure within these diverse fields generated a greater conceptual understanding, and the success of Hilbert space methods ushered in a very fruitful era for functional analysis.

Geometric intuition plays an important role in many aspects of Hilbert space theory. An element of a Hilbert space can be uniquely specified by its coordinates with respect to an orthonormal basis, in analogy with Cartesian coordinates in the plane. When that basis is countably infinite, this means that the Hilbert space can also usefully be thought of in terms of infinite sequences that are square-summable. Linear operators on a Hilbert space are likewise fairly concrete objects: in good cases, they are simply transformations that stretch the space by different factors in mutually perpendicular directions.

[edit] Motivation

Ordinary Euclidean space R³ serves as a model for the more abstract notion of a Hilbert space. In the Euclidean space, the distance between points and the angle between vectors can be expressed via the dot product, a certain bilinear operation on vectors with values in real numbers. Many problems from analytic geometry can be reworded and solved using the dot product, for example, "When are two lines orthogonal?" or "Which point on a given plane is closest to the origin?"

One of the insights of modern mathematics is that ideas from Euclidean geometry may be applied to problems which do not necessarily arise out of geometry. In a Hilbert space, the fundamental objects are abstractions of vectors, whose nature is unimportant (they may be, for example, sequences or functions of some kind). Those abstract vectors can be added and multiplied by a scalar, and an analogue of the dot product is defined for them. The algebraic operations on vectors in a Hilbert space have familiar properties, like commutativity and distributivity. In addition, the technical requirement of completeness ensures that certain limits exist. This last property is always true for finite-dimensional inner product spaces, but needs to be stated as an additional assumption in the more general case.

While the definition of a Hilbert space given below may appear complicated, due to a large number of consistency axioms, the basic intuition behind Hilbert spaces is amazingly simple:

In a large range of physical and mathematical situations, a linear problem can be stated within a certain Hilbert space and analyzed in simple geometrical terms.

In particular, this principle applies to solving linear differential and integral equations, and especially eigenvalue problems. One of the first examples of such an analysis was given by Joseph Fourier's mathematical theory of heat: a solution of the heat equation can be decomposed into infinitely many independent parts, which is closely analogous to the way of representing a vector from R³ as a linear combination of three orthogonal vectors. Similar considerations apply to other equations of mathematical physics, notably, the wave equation and Helmholtz equation.

The success of the theory of Hilbert spaces is due in part to the striking fact that

although they may differ in origin and appearance, most Hilbert spaces considered in physics and mathematics are just multiple manifestations of a single separable Hilbert space.

One way to comprehend this proceeds by introducing a system of coordinates into a given Hilbert space using the notion of orthonormal basis described below. As a consequence of the uniqueness principle, a theorem stated in abstract terms and valid in one of these spaces will hold in all of them.

[edit] Applications

Hilbert spaces allow simple geometric concepts like projection and change of basis to be extended from finite dimensional to infinite dimensional spaces, in the first place, function spaces.

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.

One goal of Fourier analysis is to write a given function as a (possibly infinite) linear combination of given basis 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 basis elements. The Fourier transform then corresponds to a change of basis.

[edit] History

The first important theorems that apply to Hilbert spaces were obtained by Joseph Fourier, Friedrich Bessel and Marc-Antoine Parseval in the 19th century in the context of periodic functions of one real variable. Fourier's theory of trigonometric series in particular provides a template for the later development of the theory of function spaces in an abstract setting. Further basic results were proved in early 20th century, for example, the Riesz representation theorem of Maurice Frechet and Frigyes Riesz from 1907.

Hilbert spaces are named after David Hilbert, who developed methods of infinite-dimensional linear algebra in the course of his work on integral equations beginning around 1909.^[1] Hilbert's axiomatic approach to the study of function spaces and operators on them, which may be termed the "algebraization of analysis", provided the foundations for functional analysis as a new mathematical discipline, and made profound impact on the later development of mathematics.

The significance of the concept of Hilbert space was underlined with the realization that it offers one of the best mathematical formulations of quantum mechanics. In short, the states of a quantum mechanical system are described by vectors in a certain Hilbert space, the observables are expressed by linear operators, and the procedure of quantum measurement is related to orthogonal projection. Moreover, the symmetries of a quantum mechanical system can be interpreted as a unitary representation of a suitable group, providing an impetus for development of unitary representation theory. On the other hand, around the same time it became clear that certain properties of classical dynamical systems can be analyzed using Hilbert space techniques in the framework of ergodic theory.

John von Neumann coined the term abstract Hilbert space in his famous work on unbounded Hermitian operators (von Neumann 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 which began in (Hilbert, Nordheim & von Neumann 1927), and continued in his work with Eugene Wigner. The name "Hilbert space" was soon adopted by others, for example by Hermann Weyl in his book on quantum mechanics and the theory of groups (Weyl 1931).

[edit] Definition and examples

A Hilbert space is a real or complex inner product space that is complete under the norm defined by the inner product $\langle\cdot,\cdot\rangle$ by^[2]

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

Some authors use slightly different definitions. For example, (Kolmogorov & Fomin 1970) define a Hilbert space as above but restrict the definition to separable infinite-dimensional spaces. A separable, infinite-dimensional Hilbert space is unique up to isomorphism; it is denoted by ℓ²(N), or simply ℓ². Older books and papers sometimes call a Hilbert space a unitary space or a linear space with an inner product, but this terminology has fallen out of use.

In the examples of Hilbert spaces given below, the underlying field of scalars is the complex numbers C, although similar definitions apply to the case in which the underlying field of scalars is the real numbers R.

[edit] Euclidean spaces

Every finite-dimensional inner product space is also a Hilbert space. For example, Cⁿ with the inner product defined by

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

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

[edit] Sequence spaces

Given a set B, the sequence space ℓ² (said "little ell two") over B is defined

$\ell^2(B) =\big\{ x : B \xrightarrow{x} \mathbb{C} \text{ and } \sum_{b \in B} \left|x \left(b\right)\right|^2 < \infty \big\}.$

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. 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 separable and is simply called ℓ².

[edit] Lebesgue spaces

Main article: Lp space

Lebesgue spaces are function spaces associated to measure spaces (X, M, μ), where X is a set, M is a σ-algebra of subsets of X, and μ is a countably additive measure on M. For example, Let L²_μ(X) be the space of those complex-valued measurable functions on X for which the Lebesgue integral of the square of the absolute value of the function is finite, and where functions are identified if and only if they differ only on a set of measure 0.

The inner product of functions f and g in L²_μ(X) is then defined as

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

This integral exists, and the resulting space is complete. See, for example, Halmos 1950, Section 42. The full Lebesgue integral is needed to ensure completeness, however, as not enough functions are Riemann integrable; see Hewitt & Stromberg 1965.

[edit] 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.

[edit] New Hilbert spaces from old

Two (or more) Hilbert spaces can be combined to produce another Hilbert space by taking either their direct sum or their tensor product.

[edit] Properties

[edit] Completeness

Completeness is the key to handling infinite-dimensional examples, such as function spaces, and is required, for instance, for the Riesz representation theorem to hold. It is expressed using a form of the Cauchy criterion for sequences in H: a normed space H is complete if every Cauchy sequence converges with respect to this norm to an element in the space.

[edit] Parallelogram identity

By definition, every Hilbert space is also a Banach space. Furthermore, in every Hilbert space the following parallelogram identity holds:

$\|u+v\|^2+\|u-v\|^2=2(\|u\|^2+\|v\|^2).$

Conversely, every Banach space in which the parallelogram identity holds is a Hilbert space, and the inner product is uniquely determined by the norm.

[edit] Topology

As for any normed vector space, an inner product space becomes a topological vector space if we declare that the open balls constitute a basis of topology.

[edit] Reflexivity

Every Hilbert space is reflexive, i.e. every Hilbert space can be naturally identified with its double dual. 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.

[edit] Orthonormal bases

A key role in the theory is played by the notion of orthonormal basis of a Hilbert space H: a family {e_k}_{k ∈ B} of H satisfying the conditions:

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

A system of vectors satisfying the first two conditions basis is called an orthonormal system or an orthonormal sequence (if B is countable). It can be proved that such a system is always linearly independent. Completeness of an orthonormal system of vectors of a Hilbert space can be equivalently restated as:

if $\langle v, e_k\rangle=0$ for all $k\in B$ and some $v\in H,$ then $v=\mathbf{0}.$

Examples of orthonormal bases include:

the set {(1,0,0),(0,1,0),(0,0,1)} forms an orthonormal basis of R³ with the dot product
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.

[edit] Hilbert dimension

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, called the Hilbert dimension of the space. In detail, 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. The cardinal number of B is the Hilbert dimension of H.

Separable spaces

A Hilbert space is separable if and only if it admits a countable orthonormal basis. All infinite-dimensional separable Hilbert spaces are isomorphic to ℓ². In particular, since all infinite-dimensional separable Hilbert spaces are isomorphic, and since almost all Hilbert spaces used in physics are infinite-dimensional and separable, when physicists talk about "the Hilbert space" or just "Hilbert space", they mean any infinite-dimensional separable one.

[edit] Orthogonal complements and projections

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

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

S^⊥ is a closed subspace of H and so forms itself a Hilbert space. If V is a closed subspace of H, then V^⊥ 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^⊥. Therefore, H is the internal Hilbert direct sum of V and V^⊥. 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.

[edit] 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.

[edit] 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 physics, 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).

[edit] See also

Wikibooks has a book on the topic of

Functional Analysis/Hilbert spaces

[edit] Notes

^ David Hilbert. Encyclopædia Britannica (2007). Retrieved on 2007-09-08.
^ The mathematical material in this article can be found in any good textbook on functional analysis, such as (Dieudonné 1960).

[edit] References

Dieudonné, Jean (1960), Foundations of Modern Analysis, Academic Press
Halmos, Paul (1950), Measure Theory, D. van Nostrand Co
Hewitt, Edwin & Stromberg, Karl (1965), Real and Abstract Analysis, Springer-Verlag
Hilbert, David; Nordheim, Lothar (Wolfgang) & von Neumann, John (1927), “Über die Grundlagen der Quantenmechanik”, Mathematische Annalen 98: 1–30, <http://dz-srv1.sub.uni-goettingen.de/sub/digbib/loader?ht=VIEW&did=D27779>
Kolmogorov, Andrey & Fomin, Sergei V. (1970), Introductory Real Analysis (Revised English edition, trans. by Richard A. Silverman (1975) ed.), Dover Press, ISBN 0-486-61226-0
B.M. Levitan (2001), “Hilbert space”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
von Neumann, John (1929), “Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren”, Mathematische Annalen 102: 49–131
Weyl, Hermann (1931), The Theory of Groups and Quantum Mechanics (English edition (1950) ed.), Dover Press, ISBN 0-486-60269-9