Hilbert space

In mathematics, a Hilbert space is a real or complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space to infinite dimensions. The inner product, which is the analog of the dot product from vector calculus, allows lengths and angles to be defined. Furthermore, completeness means that there are enough limits in the space to allow the techniques of calculus to be used. A Hilbert space is a special case of a Banach space.
Hilbert spaces were studied beginning in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis, and ergodic theory. John von Neumann coined the term Hilbert space for the abstract concept that underlies many of these diverse applications. The success of Hilbert space methods ushered in a very fruitful era for functional analysis. Apart from the classical Euclidean vector spaces, examples of Hilbert spaces include spaces of square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized functions, and Hardy spaces of holomorphic functions.
Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in a Hilbert space. At a deeper level, perpendicular projection onto a linear subspace plays a significant role in optimization problems and other aspects of the 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 classical geometry. When this basis is countably infinite, it allows identifying the Hilbert space with the space of the infinite sequences that are square-summable. The latter space is often in the older literature referred to as the Hilbert space.

Definition and illustration

Motivating example: Euclidean vector space

One of the most familiar examples of a Hilbert space is the Euclidean vector space consisting of three-dimensional vectors, denoted by, and equipped with the dot product. The dot product takes two vectors and, and produces a real number. If and are represented in Cartesian coordinates, then the dot product is defined by
The dot product satisfies the properties

It is symmetric in and :.
It is linear in its first argument: for any scalars,, and vectors,, and.
It is positive definite: for all vectors,, with equality if and only if.

An operation on pairs of vectors that, like the dot product, satisfies these three properties is known as a inner product. A vector space equipped with such an inner product is known as a inner product space. Every finite-dimensional inner product space is also a Hilbert space. The basic feature of the dot product that connects it with Euclidean geometry is that it is related to both the length of a vector, denoted, and to the angle between two vectors and by means of the formula
Multivariable calculus in Euclidean space relies on the ability to compute limits, and to have useful criteria for concluding that limits exist. A mathematical series
consisting of vectors in is absolutely convergent provided that the sum of the lengths converges as an ordinary series of real numbers:
Just as with a series of scalars, a series of vectors that converges absolutely also converges to some limit vector in the Euclidean space, in the sense that
This property expresses the completeness of Euclidean space: that a series that converges absolutely also converges in the ordinary sense.
Hilbert spaces are often taken over the complex numbers. The complex plane denoted by is equipped with a notion of magnitude, the complex modulus, which is defined as the square root of the product of with its complex conjugate:
If is a decomposition of into its real and imaginary parts, then the modulus is the usual Euclidean two-dimensional length:
The inner product of a pair of complex numbers and is the product of with the complex conjugate of :
This is complex-valued. The real part of gives the usual two-dimensional Euclidean dot product.
A second example is the space whose elements are pairs of complex numbers. Then an inner product of with another such vector is given by
The real part of is then the four-dimensional Euclidean dot product. This inner product is Hermitian symmetric, which means that the result of interchanging and is the complex conjugate:

Definition

A is a real or complex inner product space that is also a complete metric space with respect to the distance function induced by the inner product.
To say that a complex vector space is a means that there is an inner product associating a complex number to each pair of elements of that satisfies the following properties:

The inner product is conjugate symmetric; that is, the inner product of a pair of elements is equal to the complex conjugate of the inner product of the swapped elements: Importantly, this implies that is a real number.
The inner product is linear in its first argument. For all complex numbers and
The inner product of an element with itself is positive definite:

It follows from properties 1 and 2 that a complex inner product is, also called, in its second argument, meaning that
A is defined in the same way, except that is a real vector space and the inner product takes real values. Such an inner product will be a bilinear map and will form a dual system.
The norm is the real-valued function
and the distance between two points in is defined in terms of the norm by
Here, is a distance function meaning firstly that it is symmetric in and secondly that the distance between and itself is zero, and otherwise the distance between and must be positive, and lastly that the triangle inequality holds, meaning that the length of one leg of a triangle cannot exceed the sum of the lengths of the other two legs:
This last property is ultimately a consequence of the more fundamental Cauchy–Schwarz inequality, which asserts
with equality if and only if and are linearly dependent.
With a distance function defined in this way, any inner product space is a metric space, and sometimes is known as a. Any pre-Hilbert space that is additionally also a complete space is a Hilbert space.
The of is expressed using a form of the Cauchy criterion for sequences in : a pre-Hilbert space is complete if every Cauchy sequence converges with respect to this norm to an element in the space. Completeness can be characterized by the following equivalent condition: if a series of vectors
converges absolutely in the sense that
then the series converges in, in the sense that the partial sums converge to an element of.
As a complete normed space, Hilbert spaces are by definition also Banach spaces. As such they are topological vector spaces, in which topological notions like the openness and closedness of subsets are well defined. Of special importance is the notion of a closed linear subspace of a Hilbert space that, with the inner product induced by restriction, is also complete and therefore a Hilbert space in its own right.

Second example: sequence spaces

The sequence space consists of all infinite sequences of complex numbers such that the series of its squared norms converges:
The inner product on is defined by:
The series for the inner product converges as a consequence of the Cauchy–Schwarz inequality and the assumed convergence of the two series of squared norms.
Completeness of the space holds provided that whenever a series of elements from converges absolutely, then it converges to an element of. The proof is basic in mathematical analysis, and permits mathematical series of elements of the space to be manipulated with the same ease as series of complex numbers.

History

Prior to the development of Hilbert spaces, other generalizations of Euclidean spaces were known to mathematicians and physicists. In particular, the idea of an abstract linear space had gained some traction towards the end of the 19th century: this is a space whose elements can be added together and multiplied by scalars without necessarily identifying these elements with "geometric" vectors, such as position and momentum vectors in physical systems. Other objects studied by mathematicians at the turn of the 20th century, in particular spaces of sequences and spaces of functions, can naturally be thought of as linear spaces. Functions, for instance, can be added together or multiplied by constant scalars, and these operations obey the algebraic laws satisfied by addition and scalar multiplication of spatial vectors.
In the first decade of the 20th century, parallel developments led to the introduction of Hilbert spaces. The first of these was the observation, which arose during David Hilbert and Erhard Schmidt's study of integral equations, that two square-integrable real-valued functions and on an interval have an inner product
that has many of the familiar properties of the Euclidean dot product. In particular, the idea of an orthogonal family of functions has meaning. Schmidt exploited the similarity of this inner product with the usual dot product to prove an analog of the spectral decomposition for an operator of the form
where is a continuous function symmetric in and. The resulting eigenfunction expansion expresses the function as a series of the form
where the functions are orthogonal in the sense that for all. The individual terms in this series are sometimes referred to as elementary product solutions. However, there are eigenfunction expansions that fail to converge in a suitable sense to a square-integrable function: the missing ingredient, which ensures convergence, is completeness.
The second development 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 much broader class of functions. In 1907, Frigyes Riesz and Ernst Sigismund Fischer independently proved that the space of square Lebesgue-integrable functions is a 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 easily carried over 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 was independently established by Maurice Fréchet and Frigyes Riesz in 1907. John von Neumann coined the term abstract Hilbert space in his work on unbounded Hermitian operators. Although other mathematicians such as Hermann Weyl and Norbert Wiener had already studied particular Hilbert spaces in great detail, often from a physically motivated point of view, von Neumann gave the first complete and axiomatic treatment of them. Von Neumann later used them in his seminal work on the foundations of quantum mechanics, and in his continued 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 a 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 vectors in a certain Hilbert space, the observables are hermitian operators on that space, the symmetries of the system are unitary operators, and measurements are orthogonal projections. The relation between quantum mechanical symmetries and unitary operators provided an impetus for the development of the unitary representation theory of groups, initiated in the 1928 work of Hermann Weyl. On the other hand, in the early 1930s it became clear that classical mechanics can be described in terms of Hilbert space and that certain properties of classical dynamical systems can be analyzed using Hilbert space techniques in the framework of ergodic theory.
The algebra of observables in quantum mechanics is naturally an algebra of operators defined on a Hilbert space, according to Werner Heisenberg's matrix mechanics formulation of quantum theory. Von Neumann began investigating operator algebras in the 1930s, as rings of operators on a Hilbert space. The kind of algebras studied by von Neumann and his contemporaries are now known as von Neumann algebras. In the 1940s, Israel Gelfand, Mark Naimark and Irving Segal gave a definition of a kind of operator algebras called C*-algebras that on the one hand made no reference to an underlying Hilbert space, and on the other extrapolated many of the useful features of the operator algebras that had previously been studied. The spectral theorem for self-adjoint operators in particular that underlies much of the existing Hilbert space theory was generalized to C*-algebras. These techniques are now basic in abstract harmonic analysis and representation theory.