User:Silly rabbit/Hilbert space

Introduction and history

Ordinary Euclidean space R³ serves as a model for the more abstract notion of a Hilbert space. In Euclidean space, the distance between points and the angle between vectors can be expressed via the dot product, a bilinear operation on vectors whose values are real numbers. Problems from analytic geometry, such as determining whether two lines are orthogonal or finding a point on a given plane closest to the origin, can be expressed and then solved using the dot product. Another important feature of R³ is that it possesses enough structure to do calculus, because of the existence of certain limits. Hilbert spaces are a type of generalization of R³ possessing an analog of the dot product usually called an inner product, and being "complete" in a suitable sense so that calculus can be effectively performed in them.

Prior to the historical genesis of Hilbert spaces, a variety of generalizations of R³ were known to mathematicians and physicists. In particular, the idea of a linear space had gained some traction towards the end of the 19th century. This was any space in which there was a suitable way to add elements and multiply them by scalars from the real or complex numbers without worrying about whether the objects were indeed "spatial" vectors. Besides the usual spatial vectors, such as position and momenta of physical systems, other things studied by mathematicians at the turn of the 20th century could naturally be thought of as linear spaces. The chief among these were spaces of sequences (including series) and spaces of functions. Functions, for instance, can be added together or multiplied by a constant scalar, and these operations satisfy the same algebraic laws as their counterparts of addition and scalar multiplication of spatial vectors.

Several parallel developments during the first decade of the 20th century led to the introduction of Hilbert spaces. The first among these, which arose during David Hilbert and Erhard Schmidt's study of integral equations, was that one could define an inner product of two square-integrable real-valued functions on an interval [a,b] by means of the pairing

$\langle f,g\rangle =\int _{a}^{b}f(x)g(x)\,dx$

which satisfied many of the familiar properties of the Euclidean dot product. In particular, one could assign meaning to the idea of a family of functions being orthogonal or orthonormal. Schmidt exploited the similarity of this inner product with the Euclidean space to prove an analog of the spectral decomposition for an operator of the form

$f(x)\mapsto \int _{a}^{b}K(x,y)f(y)\,dy$

where K is a continuous function symmetric in x and y. The resulting eigenfunction expansion expressed the function K as a series of the form

$K(x,y)=\sum _{n}\lambda _{n}\phi _{n}(x)\phi _{n}(y)\,$

where the functions φ_n are orthogonal in the sense that 〈φ_n,φ_m〉 = 0 for all n ≠ m.

The missing ingredient was completeness: there were eigenfunction expansions which failed to converge in a suitable sense to a square-integrable function. The second development leading to the introduction of Hilbert spaces was the Lebesgue integral, an alternative to the Riemann integral introduced by Henri Lebesgue in 1904. The Lebesgue integral made it possible to integrate a great many more functions. In 1907, Frigyes Riesz and Ernst Sigismund Fischer independently proved that the space L² of square Lebesgue-integrable functions is complete metric space. As a consequence of the interplay between geometry and completeness, the 19th century results of Joseph Fourier, Friedrich Bessel and Marc-Antoine Parseval on trigonometric series were easily generalized to these more general spaces, resulting in a geometrical and analytical apparatus now usually known as the Riesz-Fischer theorem.

Further basic results were proved in the early 20th century, for example, the Riesz representation theorem of Maurice Fréchet and Frigyes Riesz from 1907. John von Neumann coined the term abstract Hilbert space in his famous work on unbounded Hermitian operators. 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, 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.

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.

Applications

Hilbert spaces allow simple geometric concepts like projection and change of basis to be extended from finite dimensional to infinite dimensional spaces such as function spaces.

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.

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 the early 20th century, for example, the Riesz representation theorem of Maurice Fréchet 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. 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 was perhaps the mathematician who most clearly recognized their importance as a result of his seminal work on the foundations of quantum mechanics, 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.

References

• Bourbaki, Nicolas (1987), Topological vector spaces, Elements of mathematics, Berlin: Springer-Verlag, ISBN 978-3540136279
• Saks, Stanisław (2005), Theory of the integral (2nd Dover ed.), Dover, ISBN 978-0486446486; originally published Monografje Matematyczne, vol. 7, Warszawa, 1937.
• Stewart, James (2006), Calculus: Concepts and Contexts (3rd ed.), Thomson/Brooks/Cole.