Английская Википедия:Hilbert space

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Файл:Standing waves on a string.gif

The state of a vibrating string can be modeled as a point in a Hilbert space. The decomposition of a vibrating string into its vibrations in distinct overtones is given by the projection of the point onto the coordinate axes in the space.

In mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric space.

The earliest Hilbert spaces were studied from this point of view 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 (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). 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 Шаблон:Math, and equipped with the dot product. The dot product takes two vectors Шаблон:Math and Шаблон:Math, and produces a real number Шаблон:Math. If Шаблон:Math and Шаблон:Math are represented in Cartesian coordinates, then the dot product is defined by <math display="block">\begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} \cdot \begin{pmatrix} y_1 \\ y_2 \\ y_3 \end{pmatrix} = x_1 y_1 + x_2 y_2 + x_3 y_3 \,.</math>

The dot product satisfies the properties

1. It is symmetric in Шаблон:Math and Шаблон:Math: Шаблон:Math.
2. It is linear in its first argument: Шаблон:Math for any scalars Шаблон:Mvar, Шаблон:Mvar, and vectors Шаблон:Math, Шаблон:Math, and Шаблон:Math.
3. It is positive definite: for all vectors Шаблон:Math, Шаблон:Math, with equality if and only if Шаблон:Math.

An operation on pairs of vectors that, like the dot product, satisfies these three properties is known as a (real) inner product. A vector space equipped with such an inner product is known as a (real) 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 (or norm) of a vector, denoted Шаблон:Math, and to the angle Шаблон:Mvar between two vectors Шаблон:Math and Шаблон:Math by means of the formula <math display="block">\mathbf{x}\cdot\mathbf{y} = \left\|\mathbf{x}\right\| \left\|\mathbf{y}\right\| \, \cos\theta \,.</math>

Файл:Completeness in Hilbert space.png

Completeness means that if a particle moves along the broken path (in blue) travelling a finite total distance, then the particle has a well-defined net displacement (in orange).

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 <math display="block">\sum_{n=0}^\infty \mathbf{x}_n</math> consisting of vectors in Шаблон:Math is absolutely convergent provided that the sum of the lengths converges as an ordinary series of real numbers: <math display="block">\sum_{k=0}^\infty \|\mathbf{x}_k\| < \infty \,.</math>

Just as with a series of scalars, a series of vectors that converges absolutely also converges to some limit vector Шаблон:Math in the Euclidean space, in the sense that <math display="block">\Biggl\| \mathbf{L} - \sum_{k=0}^N \mathbf{x}_k \Biggr\| \to 0 \quad \text{as } N \to\infty \,.</math>

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 Шаблон:Math is equipped with a notion of magnitude, the complex modulus Шаблон:Math, which is defined as the square root of the product of Шаблон:Mvar with its complex conjugate: <math display="block">|z|^2 = z\overline{z} \,.</math>

If Шаблон:Math is a decomposition of Шаблон:Mvar into its real and imaginary parts, then the modulus is the usual Euclidean two-dimensional length: <math display="block">|z| = \sqrt{x^2 + y^2} \,.</math>

The inner product of a pair of complex numbers Шаблон:Mvar and Шаблон:Mvar is the product of Шаблон:Mvar with the complex conjugate of Шаблон:Mvar: <math display="block">\langle z, w\rangle = z\overline{w}\,.</math>

This is complex-valued. The real part of Шаблон:Math gives the usual two-dimensional Euclidean dot product.

A second example is the space Шаблон:Math whose elements are pairs of complex numbers Шаблон:Math. Then the inner product of Шаблон:Mvar with another such vector Шаблон:Math is given by <math display="block">\langle z, w\rangle = z_1\overline{w_1} + z_2\overline{w_2}\,.</math>

The real part of Шаблон:Math is then the two-dimensional Euclidean dot product. This inner product is Hermitian symmetric, which means that the result of interchanging Шаблон:Mvar and Шаблон:Mvar is the complex conjugate: <math display="block">\langle w, z\rangle = \overline{\langle z, w\rangle}\,.</math>

Definition

A Шаблон:Em 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 Шаблон:Math is a Шаблон:Em means that there is an inner product <math>\langle x, y \rangle</math> associating a complex number to each pair of elements <math>x, y</math> of Шаблон:Math that satisfies the following properties:

1. 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: <math display=block> \langle y, x\rangle = \overline{\langle x, y\rangle}\,.</math> Importantly, this implies that <math>\langle x, x\rangle</math> is a real number.
2. The inner product is linear in its first argument. For all complex numbers <math>a</math> and <math>b,</math> <math display=block> \langle ax_1 + bx_2, y\rangle = a\langle x_1, y\rangle + b\langle x_2, y\rangle\,.</math>
3. The inner product of an element with itself is positive definite: <math display=block>\begin{alignat}{4}

 \langle x, x\rangle > 0 & \quad \text{ if } x \neq 0, \\
 \langle x, x\rangle = 0 & \quad \text{ if } x = 0\,.

\end{alignat}</math>

It follows from properties 1 and 2 that a complex inner product is Шаблон:Em, also called Шаблон:Em, in its second argument, meaning that <math display=block>\langle x, ay_1 + by_2\rangle = \bar{a}\langle x, y_1\rangle + \bar{b}\langle x, y_2\rangle\,.</math>

A Шаблон:Em is defined in the same way, except that Шаблон:Math is a real vector space and the inner product takes real values. Such an inner product will be a bilinear map and <math>(H, H, \langle \cdot, \cdot \rangle)</math> will form a dual system.Шаблон:Sfn

The norm is the real-valued function <math display=block>\|x\| = \sqrt{\langle x, x \rangle}\,,</math> and the distance <math>d</math> between two points <math>x, y</math> in Шаблон:Math is defined in terms of the norm by <math display=block>d(x, y) = \|x - y\| = \sqrt{\langle x - y, x - y \rangle}\,.</math>

That this function is a distance function means firstly that it is symmetric in <math>x</math> and <math>y,</math> secondly that the distance between <math>x</math> and itself is zero, and otherwise the distance between <math>x</math> and <math>y</math> must be positive, and lastly that the triangle inequality holds, meaning that the length of one leg of a triangle Шаблон:Math cannot exceed the sum of the lengths of the other two legs: <math display=block>d(x, z) \leq d(x, y) + d(y, z)\,.</math>

Файл:Triangle inequality in a metric space.svg

This last property is ultimately a consequence of the more fundamental Cauchy–Schwarz inequality, which asserts <math display=block>\left|\langle x, y\rangle\right| \leq \|x\| \|y\|</math> with equality if and only if <math>x</math> and <math>y</math> 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 Шаблон:Em. Any pre-Hilbert space that is additionally also a complete space is a Hilbert space.

The Шаблон:Em of Шаблон:Math is expressed using a form of the Cauchy criterion for sequences in Шаблон:Math: a pre-Hilbert space Шаблон:Math 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 <math display=block>\sum_{k=0}^\infty u_k</math> converges absolutely in the sense that <math display=block>\sum_{k=0}^\infty\|u_k\| < \infty\,,</math> then the series converges in Шаблон:Math, in the sense that the partial sums converge to an element of Шаблон:Math.

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 (being a closed set in a complete metric space) and therefore a Hilbert space in its own right.

Second example: sequence spaces

The sequence space Шаблон:Math consists of all infinite sequences Шаблон:Math of complex numbers such that the following series converges: <math display="block">\sum_{n=1}^\infty |z_n|^2</math>

The inner product on Шаблон:Math is defined by: <math display="block">\langle \mathbf{z}, \mathbf{w}\rangle = \sum_{n=1}^\infty z_n\overline{w_n}\,,</math>

This second series converges as a consequence of the Cauchy–Schwarz inequality and the convergence of the previous series.

Completeness of the space holds provided that whenever a series of elements from Шаблон:Math converges absolutely (in norm), then it converges to an element of Шаблон:Math. 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 (or vectors in a finite-dimensional Euclidean space).

History

Файл:Hilbert.jpg

David Hilbert

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 (vector 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 (such as real or complex numbers) 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 (including series) 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 Шаблон:Mvar and Шаблон:Mvar on an interval Шаблон:Math have an inner product

<math>\langle f, g \rangle = \int_a^b f(x)g(x)\, \mathrm{d}x</math>

which 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

<math>f(x) \mapsto \int_a^b K(x, y) f(y)\, \mathrm{d}y</math>

where Шаблон:Mvar is a continuous function symmetric in Шаблон:Mvar and Шаблон:Mvar. The resulting eigenfunction expansion expresses the function Шаблон:Mvar as a series of the form

<math>K(x, y) = \sum_n \lambda_n\varphi_n(x)\varphi_n(y)</math>

where the functions Шаблон:Mvar are orthogonal in the sense that Шаблон:Math for all Шаблон:Math. 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 Шаблон:Math 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 (Koopman–von Neumann classical mechanics) 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.

Examples

Lebesgue spaces

Lebesgue spaces are function spaces associated to measure spaces Шаблон:Math, where Шаблон:Math is a set, Шаблон:Math is a σ-algebra of subsets of Шаблон:Math, and Шаблон:Math is a countably additive measure on Шаблон:Math. Let Шаблон:Math be the space of those complex-valued measurable functions on Шаблон:Math for which the Lebesgue integral of the square of the absolute value of the function is finite, i.e., for a function Шаблон:Math in Шаблон:Math, <math display="block"> \int_X |f|^2 \mathrm{d} \mu < \infty \,, </math> and where functions are identified if and only if they differ only on a set of measure zero.

The inner product of functions Шаблон:Math and Шаблон:Math in Шаблон:Math is then defined as <math display="block">\langle f, g\rangle = \int_X f(t) \overline{g(t)} \, \mathrm{d} \mu(t) </math> or <math display="block"> \langle f, g\rangle = \int_X \overline{f(t)} g(t) \, \mathrm{d} \mu(t) \,,</math>

where the second form (conjugation of the first element) is commonly found in the theoretical physics literature. For Шаблон:Math and Шаблон:Math in Шаблон:Math, the integral exists because of the Cauchy–Schwarz inequality, and defines an inner product on the space. Equipped with this inner product, Шаблон:Math is in fact complete. The Lebesgue integral is essential to ensure completeness: on domains of real numbers, for instance, not enough functions are Riemann integrable.

The Lebesgue spaces appear in many natural settings. The spaces Шаблон:Math and Шаблон:Math of square-integrable functions with respect to the Lebesgue measure on the real line and unit interval, respectively, are natural domains on which to define the Fourier transform and Fourier series. In other situations, the measure may be something other than the ordinary Lebesgue measure on the real line. For instance, if Шаблон:Math is any positive measurable function, the space of all measurable functions Шаблон:Math on the interval Шаблон:Math satisfying <math display="block">\int_0^1 \bigl|f(t)\bigr|^2 w(t)\, \mathrm{d}t < \infty</math> is called the [[Lp space#Weighted Lp spaces|weighted Шаблон:Math space]] Шаблон:Math, and Шаблон:Math is called the weight function. The inner product is defined by <math display="block">\langle f, g\rangle = \int_0^1 f(t) \overline{g(t)} w(t) \, \mathrm{d}t \,.</math>

The weighted space Шаблон:Math is identical with the Hilbert space Шаблон:Math where the measure Шаблон:Math of a Lebesgue-measurable set Шаблон:Math is defined by <math display="block">\mu(A) = \int_A w(t)\,\mathrm{d}t \,.</math>

Weighted Шаблон:Math spaces like this are frequently used to study orthogonal polynomials, because different families of orthogonal polynomials are orthogonal with respect to different weighting functions.

Sobolev spaces

Sobolev spaces, denoted by Шаблон:Math or Шаблон:Math, are Hilbert spaces. These are a special kind of function space in which differentiation may be performed, but that (unlike other Banach spaces such as the Hölder spaces) support the structure of an inner product. Because differentiation is permitted, Sobolev spaces are a convenient setting for the theory of partial differential equations. They also form the basis of the theory of direct methods in the calculus of variations.

For Шаблон:Math a non-negative integer and Шаблон:Math, the Sobolev space Шаблон:Math contains Шаблон:Math functions whose weak derivatives of order up to Шаблон:Math are also Шаблон:Math. The inner product in Шаблон:Math is <math display="block">\langle f, g\rangle = \int_\Omega f(x)\bar{g}(x)\,\mathrm{d}x + \int_\Omega D f(x)\cdot D\bar{g}(x)\,\mathrm{d}x + \cdots + \int_\Omega D^s f(x)\cdot D^s \bar{g}(x)\, \mathrm{d}x</math> where the dot indicates the dot product in the Euclidean space of partial derivatives of each order. Sobolev spaces can also be defined when Шаблон:Math is not an integer.

Sobolev spaces are also studied from the point of view of spectral theory, relying more specifically on the Hilbert space structure. If Шаблон:Math is a suitable domain, then one can define the Sobolev space Шаблон:Math as the space of Bessel potentials; roughly, <math display="block">H^s(\Omega) = \left\{ (1-\Delta)^{-s/2}f \mathrel{\Big|} f\in L^2(\Omega)\right\} \,.</math>

Here Шаблон:Math is the Laplacian and Шаблон:Math is understood in terms of the spectral mapping theorem. Apart from providing a workable definition of Sobolev spaces for non-integer Шаблон:Math, this definition also has particularly desirable properties under the Fourier transform that make it ideal for the study of pseudodifferential operators. Using these methods on a compact Riemannian manifold, one can obtain for instance the Hodge decomposition, which is the basis of Hodge theory.

Spaces of holomorphic functions

Hardy spaces

The Hardy spaces are function spaces, arising in complex analysis and harmonic analysis, whose elements are certain holomorphic functions in a complex domain. Let Шаблон:Math denote the unit disc in the complex plane. Then the Hardy space Шаблон:Math is defined as the space of holomorphic functions Шаблон:Math on Шаблон:Math such that the means

<math display="block">M_r(f) = \frac{1}{2\pi} \int_0^{2\pi} \left|f\bigl(re^{i\theta}\bigr)\right|^2 \, \mathrm{d}\theta</math>

remain bounded for Шаблон:Math. The norm on this Hardy space is defined by <math display="block">\left\|f\right\|_2 = \lim_{r \to 1} \sqrt{M_r(f)} \,.</math>

Hardy spaces in the disc are related to Fourier series. A function Шаблон:Math is in Шаблон:Math if and only if <math display="block">f(z) = \sum_{n=0}^\infty a_n z^n</math> where <math display="block">\sum_{n=0}^\infty |a_n|^2 < \infty \,.</math>

Thus Шаблон:Math consists of those functions that are L² on the circle, and whose negative frequency Fourier coefficients vanish.

Bergman spaces

The Bergman spaces are another family of Hilbert spaces of holomorphic functions. Let Шаблон:Math be a bounded open set in the complex plane (or a higher-dimensional complex space) and let Шаблон:Math be the space of holomorphic functions Шаблон:Math in Шаблон:Math that are also in Шаблон:Math in the sense that <math display="block">\|f\|^2 = \int_D |f(z)|^2\,\mathrm{d}\mu(z) < \infty \,,</math>

where the integral is taken with respect to the Lebesgue measure in Шаблон:Math. Clearly Шаблон:Math is a subspace of Шаблон:Math; in fact, it is a closed subspace, and so a Hilbert space in its own right. This is a consequence of the estimate, valid on compact subsets Шаблон:Math of Шаблон:Math, that <math display="block">\sup_{z\in K} \left|f(z)\right| \le C_K \left\|f\right\|_2 \,,</math> which in turn follows from Cauchy's integral formula. Thus convergence of a sequence of holomorphic functions in Шаблон:Math implies also compact convergence, and so the limit function is also holomorphic. Another consequence of this inequality is that the linear functional that evaluates a function Шаблон:Math at a point of Шаблон:Math is actually continuous on Шаблон:Math. The Riesz representation theorem implies that the evaluation functional can be represented as an element of Шаблон:Math. Thus, for every Шаблон:Math, there is a function Шаблон:Math such that <math display="block">f(z) = \int_D f(\zeta)\overline{\eta_z(\zeta)}\,\mathrm{d}\mu(\zeta)</math> for all Шаблон:Math. The integrand <math display="block">K(\zeta, z) = \overline{\eta_z(\zeta)}</math> is known as the Bergman kernel of Шаблон:Math. This integral kernel satisfies a reproducing property <math display="block">f(z) = \int_D f(\zeta)K(\zeta, z)\,\mathrm{d}\mu(\zeta) \,.</math>

A Bergman space is an example of a reproducing kernel Hilbert space, which is a Hilbert space of functions along with a kernel Шаблон:Math that verifies a reproducing property analogous to this one. The Hardy space Шаблон:Math also admits a reproducing kernel, known as the Szegő kernel. Reproducing kernels are common in other areas of mathematics as well. For instance, in harmonic analysis the Poisson kernel is a reproducing kernel for the Hilbert space of square-integrable harmonic functions in the unit ball. That the latter is a Hilbert space at all is a consequence of the mean value theorem for harmonic functions.

Applications

Many of the applications of Hilbert spaces exploit the fact that Hilbert spaces support generalizations of simple geometric concepts like projection and change of basis from their usual finite dimensional setting. In particular, the spectral theory of continuous self-adjoint linear operators on a Hilbert space generalizes the usual spectral decomposition of a matrix, and this often plays a major role in applications of the theory to other areas of mathematics and physics.

Sturm–Liouville theory

Файл:Harmonic partials on strings.svg

The overtones of a vibrating string. These are eigenfunctions of an associated Sturm–Liouville problem. The eigenvalues 1, Шаблон:Sfrac, Шаблон:Sfrac, ... form the (musical) harmonic series.

In the theory of ordinary differential equations, spectral methods on a suitable Hilbert space are used to study the behavior of eigenvalues and eigenfunctions of differential equations. For example, the Sturm–Liouville problem arises in the study of the harmonics of waves in a violin string or a drum, and is a central problem in ordinary differential equations. The problem is a differential equation of the form <math display="block"> -\frac{\mathrm{d}}{\mathrm{d}x}\left[p(x)\frac{\mathrm{d}y}{\mathrm{d}x}\right] + q(x)y = \lambda w(x)y</math> for an unknown function Шаблон:Math on an interval Шаблон:Closed-closed, satisfying general homogeneous Robin boundary conditions <math display="block">\begin{cases}

 \alpha y(a)+\alpha' y'(a) &= 0 \\
 \beta y(b) + \beta' y'(b) &= 0 \,.

\end{cases}</math> The functions Шаблон:Math, Шаблон:Math, and Шаблон:Math are given in advance, and the problem is to find the function Шаблон:Math and constants Шаблон:Math for which the equation has a solution. The problem only has solutions for certain values of Шаблон:Math, called eigenvalues of the system, and this is a consequence of the spectral theorem for compact operators applied to the integral operator defined by the Green's function for the system. Furthermore, another consequence of this general result is that the eigenvalues Шаблон:Math of the system can be arranged in an increasing sequence tending to infinity.

Partial differential equations

Hilbert spaces form a basic tool in the study of partial differential equations. For many classes of partial differential equations, such as linear elliptic equations, it is possible to consider a generalized solution (known as a weak solution) by enlarging the class of functions. Many weak formulations involve the class of Sobolev functions, which is a Hilbert space. A suitable weak formulation reduces to a geometrical problem, the analytic problem of finding a solution or, often what is more important, showing that a solution exists and is unique for given boundary data. For linear elliptic equations, one geometrical result that ensures unique solvability for a large class of problems is the Lax–Milgram theorem. This strategy forms the rudiment of the Galerkin method (a finite element method) for numerical solution of partial differential equations.

A typical example is the Poisson equation Шаблон:Math with Dirichlet boundary conditions in a bounded domain Шаблон:Math in Шаблон:Math. The weak formulation consists of finding a function Шаблон:Math such that, for all continuously differentiable functions Шаблон:Math in Шаблон:Math vanishing on the boundary: <math display="block">\int_\Omega \nabla u\cdot\nabla v = \int_\Omega gv\,.</math>

This can be recast in terms of the Hilbert space Шаблон:Math consisting of functions Шаблон:Math such that Шаблон:Math, along with its weak partial derivatives, are square integrable on Шаблон:Math, and vanish on the boundary. The question then reduces to finding Шаблон:Math in this space such that for all Шаблон:Math in this space <math display="block">a(u, v) = b(v)</math>

where Шаблон:Math is a continuous bilinear form, and Шаблон:Math is a continuous linear functional, given respectively by <math display="block">a(u, v) = \int_\Omega \nabla u\cdot\nabla v,\quad b(v)= \int_\Omega gv\,.</math>

Since the Poisson equation is elliptic, it follows from Poincaré's inequality that the bilinear form Шаблон:Math is coercive. The Lax–Milgram theorem then ensures the existence and uniqueness of solutions of this equation.

Hilbert spaces allow for many elliptic partial differential equations to be formulated in a similar way, and the Lax–Milgram theorem is then a basic tool in their analysis. With suitable modifications, similar techniques can be applied to parabolic partial differential equations and certain hyperbolic partial differential equations.

Ergodic theory

Файл:BunimovichStadium.svg

The path of a billiard ball in the Bunimovich stadium is described by an ergodic dynamical system.

The field of ergodic theory is the study of the long-term behavior of chaotic dynamical systems. The protypical case of a field that ergodic theory applies to is thermodynamics, in which—though the microscopic state of a system is extremely complicated (it is impossible to understand the ensemble of individual collisions between particles of matter)—the average behavior over sufficiently long time intervals is tractable. The laws of thermodynamics are assertions about such average behavior. In particular, one formulation of the zeroth law of thermodynamics asserts that over sufficiently long timescales, the only functionally independent measurement that one can make of a thermodynamic system in equilibrium is its total energy, in the form of temperature.

An ergodic dynamical system is one for which, apart from the energy—measured by the Hamiltonian—there are no other functionally independent conserved quantities on the phase space. More explicitly, suppose that the energy Шаблон:Math is fixed, and let Шаблон:Math be the subset of the phase space consisting of all states of energy Шаблон:Math (an energy surface), and let Шаблон:Math denote the evolution operator on the phase space. The dynamical system is ergodic if every invariant measurable functions on Шаблон:Math is constant almost everywhere. An invariant function Шаблон:Math is one for which <math display="block">f(T_tw) = f(w)</math> for all Шаблон:Math on Шаблон:Math and all time Шаблон:Math. Liouville's theorem implies that there exists a measure Шаблон:Math on the energy surface that is invariant under the time translation. As a result, time translation is a unitary transformation of the Hilbert space Шаблон:Math consisting of square-integrable functions on the energy surface Шаблон:Math with respect to the inner product <math display="block">\left\langle f, g\right\rangle_{L^2\left(\Omega_E, \mu\right)} = \int_E f\bar{g}\,\mathrm{d}\mu\,.</math>

The von Neumann mean ergodic theorem states the following:

• If Шаблон:Math is a (strongly continuous) one-parameter semigroup of unitary operators on a Hilbert space Шаблон:Math, and Шаблон:Math is the orthogonal projection onto the space of common fixed points of Шаблон:Math, Шаблон:Math, then <math display="block"> Px = \lim_{T\to\infty} \frac{1}{T} \int_0^T U_tx\,\mathrm{d}t\,.</math>

For an ergodic system, the fixed set of the time evolution consists only of the constant functions, so the ergodic theorem implies the following: for any function Шаблон:Math, <math display="block">\underset{T\to\infty}{L^2 - \lim} \frac{1}{T}\int_0^T f(T_tw)\,\mathrm{d}t = \int_{\Omega_E} f(y)\,\mathrm{d}\mu(y)\,.</math>

That is, the long time average of an observable Шаблон:Math is equal to its expectation value over an energy surface.

Fourier analysis

Файл:Sawtooth Fourier Analysys.svg

Superposition of sinusoidal wave basis functions (bottom) to form a sawtooth wave (top)

Файл:Harmoniki.png

Spherical harmonics, an orthonormal basis for the Hilbert space of square-integrable functions on the sphere, shown graphed along the radial direction

One of the basic goals of Fourier analysis is to decompose a function into a (possibly infinite) linear combination of given basis functions: the associated Fourier series. The classical Fourier series associated to a function Шаблон:Math defined on the interval Шаблон:Math is a series of the form <math display="block">\sum_{n=-\infty}^\infty a_n e^{2\pi in\theta}</math> where <math display="block">a_n = \int_0^1f(\theta)\;\!e^{-2\pi in\theta}\,\mathrm{d}\theta\,.</math>

The example of adding up the first few terms in a Fourier series for a sawtooth function is shown in the figure. The basis functions are sine waves with wavelengths Шаблон:Math (for integer Шаблон:Math) shorter than the wavelength Шаблон:Math of the sawtooth itself (except for Шаблон:Math, the fundamental wave).

A significant problem in classical Fourier series asks in what sense the Fourier series converges, if at all, to the function Шаблон:Math. Hilbert space methods provide one possible answer to this question. The functions Шаблон:Math form an orthogonal basis of the Hilbert space Шаблон:Math. Consequently, any square-integrable function can be expressed as a series <math display="block">f(\theta) = \sum_n a_n e_n(\theta)\,,\quad a_n = \langle f, e_n\rangle</math>

and, moreover, this series converges in the Hilbert space sense (that is, in the [[mean convergence|Шаблон:Math mean]]).

The problem can also be studied from the abstract point of view: 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 coefficients appearing on these basis elements are sometimes known abstractly as the Fourier coefficients of the element of the space. The abstraction is especially useful when it is more natural to use different basis functions for a space such as Шаблон:Math. In many circumstances, it is desirable not to decompose a function into trigonometric functions, but rather into orthogonal polynomials or wavelets for instance, and in higher dimensions into spherical harmonics.

For instance, if Шаблон:Math are any orthonormal basis functions of Шаблон:Math, then a given function in Шаблон:Math can be approximated as a finite linear combination <math display="block">f(x) \approx f_n (x) = a_1 e_1 (x) + a_2 e_2(x) + \cdots + a_n e_n (x)\,.</math>

The coefficients Шаблон:Math are selected to make the magnitude of the difference Шаблон:Math as small as possible. Geometrically, the best approximation is the orthogonal projection of Шаблон:Math onto the subspace consisting of all linear combinations of the Шаблон:Math, and can be calculated by <math display="block">a_j = \int_0^1 \overline{e_j(x)}f (x) \, \mathrm{d}x\,.</math>

That this formula minimizes the difference Шаблон:Math is a consequence of Bessel's inequality and Parseval's formula.

In various applications to physical problems, a function can be decomposed into physically meaningful eigenfunctions of a differential operator (typically the Laplace operator): this forms the foundation for the spectral study of functions, in reference to the spectrum of the differential operator. A concrete physical application involves the problem of hearing the shape of a drum: given the fundamental modes of vibration that a drumhead is capable of producing, can one infer the shape of the drum itself? The mathematical formulation of this question involves the Dirichlet eigenvalues of the Laplace equation in the plane, that represent the fundamental modes of vibration in direct analogy with the integers that represent the fundamental modes of vibration of the violin string.

Spectral theory also underlies certain aspects of the Fourier transform of a function. Whereas Fourier analysis decomposes a function defined on a compact set into the discrete spectrum of the Laplacian (which corresponds to the vibrations of a violin string or drum), the Fourier transform of a function is the decomposition of a function defined on all of Euclidean space into its components in the continuous spectrum of the Laplacian. The Fourier transformation is also geometrical, in a sense made precise by the Plancherel theorem, that asserts that it is an isometry of one Hilbert space (the "time domain") with another (the "frequency domain"). This isometry property of the Fourier transformation is a recurring theme in abstract harmonic analysis (since it reflects the conservation of energy for the continuous Fourier Transform), as evidenced for instance by the Plancherel theorem for spherical functions occurring in noncommutative harmonic analysis.

Quantum mechanics

Файл:HAtomOrbitals.png

The orbitals of an electron in a hydrogen atom are eigenfunctions of the energy.

Шаблон:Main In the mathematically rigorous formulation of quantum mechanics, developed by John von Neumann, the possible states (more precisely, the pure states) of a quantum mechanical system are represented by unit vectors (called state vectors) residing in a complex separable Hilbert space, known as the state space, well defined up to a complex number of norm 1 (the phase factor). In other words, the possible states are points in the projectivization of a Hilbert space, usually called the complex projective space. The exact nature of this Hilbert space is dependent on the system; for example, the position and momentum states for a single non-relativistic spin zero particle is the space of all square-integrable functions, while the states for the spin of a single proton are unit elements of the two-dimensional complex Hilbert space of spinors. Each observable is represented by a self-adjoint linear operator acting on the state space. Each eigenstate of an observable corresponds to an eigenvector of the operator, and the associated eigenvalue corresponds to the value of the observable in that eigenstate.

The inner product between two state vectors is a complex number known as a probability amplitude. During an ideal measurement of a quantum mechanical system, the probability that a system collapses from a given initial state to a particular eigenstate is given by the square of the absolute value of the probability amplitudes between the initial and final states. The possible results of a measurement are the eigenvalues of the operator—which explains the choice of self-adjoint operators, for all the eigenvalues must be real. The probability distribution of an observable in a given state can be found by computing the spectral decomposition of the corresponding operator.

For a general system, states are typically not pure, but instead are represented as statistical mixtures of pure states, or mixed states, given by density matrices: self-adjoint operators of trace one on a Hilbert space. Moreover, for general quantum mechanical systems, the effects of a single measurement can influence other parts of a system in a manner that is described instead by a positive operator valued measure. Thus the structure both of the states and observables in the general theory is considerably more complicated than the idealization for pure states.

Probability theory

In probability theory, Hilbert spaces also have diverse applications. Here a fundamental Hilbert space is the space of random variables on a given probability space, having class <math>L^2</math> (finite first and second moments). A common operation in statistics is that of centering a random variable by subtracting its expectation. Thus if <math>X</math> is a random variable, then <math>X - E(X)</math> is its centering. In the Hilbert space view, this is the orthogonal projection of <math>X</math> onto the kernel of the expectation operator, which a continuous linear functional on the Hilbert space (in fact, the inner product with the constant random variable 1), and so this kernel is a closed subspace.

The conditional expectation has a natural interpretation in the Hilbert space. Suppose that a probability space <math>(\Omega, P, \mathcal B)</math> is given, where <math>\mathcal B</math> is a sigma algebra on the set <math>\Omega</math>, and <math>P</math> is a probability measure on the measure space <math>(\Omega, \mathcal B)</math>. If <math>\mathcal F\le\mathcal B</math> is a sigma subalgebra of <math>\mathcal B</math>, then the conditional expectation <math>E[X|\mathcal F]</math> is the orthogonal projection of <math>X</math> onto the subspace of <math>L^2(\Omega, P)</math> consisting of the <math>\mathcal F</math>-measurable functions. If the random variable <math>X</math> in <math>L^2(\Omega, P)</math> is independent of the sigma algebra <math>\mathcal F</math> then conditional expectation <math>E(X|\mathcal F) = E(X)</math>, i.e., its projection onto the <math>\mathcal F</math>-measurable functions is constant. Equivalently, the projection of its centering is zero.

In particular, if two random variables <math>X</math> and <math>Y</math> (in <math>L^2(\Omega, P)</math>) are independent, then the centered random variables <math>X-E(X)</math> and <math>Y-E(Y)</math> are orthogonal. (This means that the two variables have zero covariance: they are uncorrelated.) In that case, the Pythagorean theorem in the kernel of the expectation operator implies that the variances of <math>X</math> and <math>Y</math> satisfy the identity: <math display="block">\operatorname{Var}(X+Y) = \operatorname{Var}(X) + \operatorname{Var}(Y),</math> sometimes called the Pythagorean theorem of statistics, and is of importance in linear regression. As Шаблон:Harvtxt puts it, "the analysis of variance may be viewed as the decomposition of the squared length of a vector into the sum of the squared lengths of several vectors, using the Pythagorean Theorem."

The theory of martingales can be formulated in Hilbert spaces. A martingale in a Hilbert space is a sequence <math>x_1,x_2,\dots</math> of elements of a Hilbert space such that, for each Шаблон:Math, <math>x_n</math> is the orthogonal projection of <math>x_{n+1}</math> onto the linear hull of <math>x_1,\dots,x_n</math>. If the <math>x_k</math> are random variables, this reproduces the usual definition of a (discrete) martingale: the expectation of <math>x_{n+1}</math>, conditioned on <math>x_1,\dots,x_n</math>, is equal to <math>x_n</math>.

Hilbert spaces are also used throughout the foundations of the Itô calculus. To any square-integrable martingale, it is possible to associate a Hilbert norm on the space of equivalence classes of progressively measurable processes with respect to the martingale (using the quadratic variation of the martingale as the measure). The Itô integral can be constructed by first defining it for simple processes, and then exploiting their density in the Hilbert space. A noteworthy result is then the Itô isometry, which attests that for any martingale M having quadratic variation measure <math>d\langle M\rangle_t</math>, and any progressively measurable process H: <math display="block">E\left[\left(\int_0^tH_sdM_s\right)^2\right] = E\left[\int_0^tH_s^2d\langle M\rangle_s\right]</math> whenever the expectation on the right-hand side is finite.

A deeper application of Hilbert spaces that is especially important in the theory of Gaussian processes is an attempt, due to Leonard Gross and others, to make sense of certain formal integrals over infinite dimensional spaces like the Feynman path integral from quantum field theory. The problem with integral like this is that there is no infinite dimensional Lebesgue measure. The notion of an abstract Wiener space allows one to construct a measure on a Banach space Шаблон:Math that contains a Hilbert space Шаблон:Math, called the Cameron–Martin space, as a dense subset, out of a finitely additive cylinder set measure on Шаблон:Math. The resulting measure on Шаблон:Math is countably additive and invariant under translation by elements of Шаблон:Math, and this provides a mathematically rigorous way of thinking of the Wiener measure as a Gaussian measure on the Sobolev space <math>H^1([0,\infty))</math>.

Color perception

Шаблон:Main Any true physical color can be represented by a combination of pure spectral colors. As physical colors can be composed of any number of spectral colors, the space of physical colors may aptly be represented by a Hilbert space over spectral colors. Humans have three types of cone cells for color perception, so the perceivable colors can be represented by 3-dimensional Euclidean space. The many-to-one linear mapping from the Hilbert space of physical colors to the Euclidean space of human perceivable colors explains why many distinct physical colors may be perceived by humans to be identical (e.g., pure yellow light versus a mix of red and green light, see metamerism).

Properties

Pythagorean identity

Two vectors Шаблон:Math and Шаблон:Math in a Hilbert space Шаблон:Math are orthogonal when Шаблон:Math. The notation for this is Шаблон:Math. More generally, when Шаблон:Math is a subset in Шаблон:Math, the notation Шаблон:Math means that Шаблон:Math is orthogonal to every element from Шаблон:Math.

When Шаблон:Math and Шаблон:Math are orthogonal, one has <math display="block">\|u + v\|^2 = \langle u + v, u + v \rangle = \langle u, u \rangle + 2 \, \operatorname{Re} \langle u, v \rangle + \langle v, v \rangle= \|u\|^2 + \|v\|^2\,.</math>

By induction on Шаблон:Math, this is extended to any family Шаблон:Math of Шаблон:Math orthogonal vectors, <math display="block">\left\|u_1 + \cdots + u_n\right\|^2 = \left\|u_1\right\|^2 + \cdots + \left\|u_n\right\|^2 .</math>

Whereas the Pythagorean identity as stated is valid in any inner product space, completeness is required for the extension of the Pythagorean identity to series. A series Шаблон:Math of orthogonal vectors converges in Шаблон:Math if and only if the series of squares of norms converges, and <math display="block">\Biggl\|\sum_{k=0}^\infty u_k \Biggr\|^2 = \sum_{k=0}^\infty \left\|u_k\right\|^2\,.</math> Furthermore, the sum of a series of orthogonal vectors is independent of the order in which it is taken.

Parallelogram identity and polarization

Файл:Color parallelogram.svg

Geometrically, the parallelogram identity asserts that Шаблон:Math. In words, the sum of the squares of the diagonals is twice the sum of the squares of any two adjacent sides.

By definition, every Hilbert space is also a Banach space. Furthermore, in every Hilbert space the following parallelogram identity holds: <math display="block">\|u + v\|^2 + \|u - v\|^2 = 2\bigl(\|u\|^2 + \|v\|^2\bigr)\,.</math>

Conversely, every Banach space in which the parallelogram identity holds is a Hilbert space, and the inner product is uniquely determined by the norm by the polarization identity. For real Hilbert spaces, the polarization identity is <math display="block">\langle u, v\rangle = \tfrac{1}{4}\bigl(\|u + v\|^2 - \|u - v\|^2\bigr)\,.</math>

For complex Hilbert spaces, it is <math display="block">\langle u, v\rangle = \tfrac{1}{4}\bigl(\|u + v\|^2 - \|u - v\|^2 + i\|u + iv\|^2 - i\|u - iv\|^2\bigr)\,.</math>

The parallelogram law implies that any Hilbert space is a uniformly convex Banach space.

Best approximation

This subsection employs the Hilbert projection theorem. If Шаблон:Math is a non-empty closed convex subset of a Hilbert space Шаблон:Math and Шаблон:Math a point in Шаблон:Math, there exists a unique point Шаблон:Math that minimizes the distance between Шаблон:Math and points in Шаблон:Math, <math display="block"> y \in C \,, \quad \|x - y\| = \operatorname{dist}(x, C) = \min \bigl\{ \|x - z\| \mathrel{\big|} z \in C \bigr\}\,.</math>

This is equivalent to saying that there is a point with minimal norm in the translated convex set Шаблон:Math. The proof consists in showing that every minimizing sequence Шаблон:Math is Cauchy (using the parallelogram identity) hence converges (using completeness) to a point in Шаблон:Math that has minimal norm. More generally, this holds in any uniformly convex Banach space.

When this result is applied to a closed subspace Шаблон:Math of Шаблон:Math, it can be shown that the point Шаблон:Math closest to Шаблон:Math is characterized by <math display="block"> y \in F \,, \quad x - y \perp F \,.</math>

This point Шаблон:Math is the orthogonal projection of Шаблон:Math onto Шаблон:Math, and the mapping Шаблон:Math is linear (see Orthogonal complements and projections). This result is especially significant in applied mathematics, especially numerical analysis, where it forms the basis of least squares methods.

In particular, when Шаблон:Math is not equal to Шаблон:Math, one can find a nonzero vector Шаблон:Math orthogonal to Шаблон:Math (select Шаблон:Math and Шаблон:Math). A very useful criterion is obtained by applying this observation to the closed subspace Шаблон:Math generated by a subset Шаблон:Math of Шаблон:Math.

A subset Шаблон:Math of Шаблон:Math spans a dense vector subspace if (and only if) the vector 0 is the sole vector Шаблон:Math orthogonal to Шаблон:Math.

Duality

The dual space Шаблон:Math is the space of all continuous linear functions from the space Шаблон:Math into the base field. It carries a natural norm, defined by <math display="block">\|\varphi\| = \sup_{\|x\|=1, x\in H} |\varphi(x)| \,.</math> This norm satisfies the parallelogram law, and so the dual space is also an inner product space where this inner product can be defined in terms of this dual norm by using the polarization identity. The dual space is also complete so it is a Hilbert space in its own right. If Шаблон:Math is a complete orthonormal basis for Шаблон:Mvar then the inner product on the dual space of any two <math>f, g \in H^*</math> is <math display="block">\langle f, g \rangle_{H^{*}} = \sum_{i \in I} f (e_i) \overline{g (e_i)}</math> where all but countably many of the terms in this series are zero.

The Riesz representation theorem affords a convenient description of the dual space. To every element Шаблон:Math of Шаблон:Math, there is a unique element Шаблон:Math of Шаблон:Math, defined by <math display="block">\varphi_u(x) = \langle x, u\rangle </math> where moreover, <math>\left\| \varphi_u \right\| = \left\| u \right\|.</math>

The Riesz representation theorem states that the map from Шаблон:Math to Шаблон:Math defined by Шаблон:Math is surjective, which makes this map an isometric antilinear isomorphism. So to every element Шаблон:Math of the dual Шаблон:Math there exists one and only one Шаблон:Math in Шаблон:Math such that <math display="block">\langle x, u_\varphi\rangle = \varphi(x)</math> for all Шаблон:Math. The inner product on the dual space Шаблон:Math satisfies <math display="block"> \langle \varphi, \psi \rangle = \langle u_\psi, u_\varphi \rangle \,.</math>

The reversal of order on the right-hand side restores linearity in Шаблон:Math from the antilinearity of Шаблон:Math. In the real case, the antilinear isomorphism from Шаблон:Math to its dual is actually an isomorphism, and so real Hilbert spaces are naturally isomorphic to their own duals.

The representing vector Шаблон:Math is obtained in the following way. When Шаблон:Math, the kernel Шаблон:Math is a closed vector subspace of Шаблон:Math, not equal to Шаблон:Math, hence there exists a nonzero vector Шаблон:Math orthogonal to Шаблон:Math. The vector Шаблон:Math is a suitable scalar multiple Шаблон:Math of Шаблон:Math. The requirement that Шаблон:Math yields <math display="block"> u = \langle v, v \rangle^{-1} \, \overline{\varphi (v)} \, v \,.</math>

This correspondence Шаблон:Math is exploited by the bra–ket notation popular in physics. It is common in physics to assume that the inner product, denoted by Шаблон:Math, is linear on the right, <math display="block">\langle x | y \rangle = \langle y, x \rangle \,.</math> The result Шаблон:Math can be seen as the action of the linear functional Шаблон:Math (the bra) on the vector Шаблон:Math (the ket).

The Riesz representation theorem relies fundamentally not just on the presence of an inner product, but also on the completeness of the space. In fact, the theorem implies that the topological dual of any inner product space can be identified with its completion. An immediate consequence of the Riesz representation theorem is also that a Hilbert space Шаблон:Math is reflexive, meaning that the natural map from Шаблон:Math into its double dual space is an isomorphism.

Weakly-convergent sequences

Шаблон:Main In a Hilbert space Шаблон:Math, a sequence Шаблон:Math is weakly convergent to a vector Шаблон:Math when <math display="block">\lim_n \langle x_n, v \rangle = \langle x, v \rangle</math> for every Шаблон:Math.

For example, any orthonormal sequence Шаблон:Math converges weakly to 0, as a consequence of Bessel's inequality. Every weakly convergent sequence Шаблон:Math is bounded, by the uniform boundedness principle.

Conversely, every bounded sequence in a Hilbert space admits weakly convergent subsequences (Alaoglu's theorem). This fact may be used to prove minimization results for continuous convex functionals, in the same way that the Bolzano–Weierstrass theorem is used for continuous functions on Шаблон:Math. Among several variants, one simple statement is as follows:

If Шаблон:Math is a convex continuous function such that Шаблон:Math tends to Шаблон:Math when Шаблон:Math tends to Шаблон:Math, then Шаблон:Math admits a minimum at some point Шаблон:Math.

This fact (and its various generalizations) are fundamental for direct methods in the calculus of variations. Minimization results for convex functionals are also a direct consequence of the slightly more abstract fact that closed bounded convex subsets in a Hilbert space Шаблон:Math are weakly compact, since Шаблон:Math is reflexive. The existence of weakly convergent subsequences is a special case of the Eberlein–Šmulian theorem.

Banach space properties

Any general property of Banach spaces continues to hold for Hilbert spaces. The open mapping theorem states that a continuous surjective linear transformation from one Banach space to another is an open mapping meaning that it sends open sets to open sets. A corollary is the bounded inverse theorem, that a continuous and bijective linear function from one Banach space to another is an isomorphism (that is, a continuous linear map whose inverse is also continuous). This theorem is considerably simpler to prove in the case of Hilbert spaces than in general Banach spaces. The open mapping theorem is equivalent to the closed graph theorem, which asserts that a linear function from one Banach space to another is continuous if and only if its graph is a closed set. In the case of Hilbert spaces, this is basic in the study of unbounded operators (see closed operator).

The (geometrical) Hahn–Banach theorem asserts that a closed convex set can be separated from any point outside it by means of a hyperplane of the Hilbert space. This is an immediate consequence of the best approximation property: if Шаблон:Math is the element of a closed convex set Шаблон:Math closest to Шаблон:Math, then the separating hyperplane is the plane perpendicular to the segment Шаблон:Math passing through its midpoint.

Operators on Hilbert spaces

Bounded operators

The continuous linear operators Шаблон:Math from a Hilbert space Шаблон:Math to a second Hilbert space Шаблон:Math are bounded in the sense that they map bounded sets to bounded sets. Conversely, if an operator is bounded, then it is continuous. The space of such bounded linear operators has a norm, the operator norm given by <math display="block">\lVert A \rVert = \sup \bigl\{\| Ax \| \mathrel{\big|} \| x \| \leq 1 \bigr\}\,.</math>

The sum and the composite of two bounded linear operators is again bounded and linear. For y in H₂, the map that sends Шаблон:Math to Шаблон:Math is linear and continuous, and according to the Riesz representation theorem can therefore be represented in the form <math display="block">\left\langle x, A^* y \right\rangle = \langle Ax, y \rangle</math> for some vector Шаблон:Math in Шаблон:Math. This defines another bounded linear operator Шаблон:Math, the adjoint of Шаблон:Mvar. The adjoint satisfies Шаблон:Math. When the Riesz representation theorem is used to identify each Hilbert space with its continuous dual space, the adjoint of Шаблон:Mvar can be shown to be identical to the transpose Шаблон:Math of Шаблон:Mvar, which by definition sends <math>\psi \in H_2^{*}</math> to the functional <math>\psi \circ A \in H_1^{*}.</math>

The set Шаблон:Math of all bounded linear operators on Шаблон:Math (meaning operators Шаблон:Math), together with the addition and composition operations, the norm and the adjoint operation, is a C*-algebra, which is a type of operator algebra.

An element Шаблон:Math of Шаблон:Math is called 'self-adjoint' or 'Hermitian' if Шаблон:Math. If Шаблон:Math is Hermitian and Шаблон:Math for every Шаблон:Math, then Шаблон:Math is called 'nonnegative', written Шаблон:Math; if equality holds only when Шаблон:Math, then Шаблон:Math is called 'positive'. The set of self adjoint operators admits a partial order, in which Шаблон:Math if Шаблон:Math. If Шаблон:Math has the form Шаблон:Math for some Шаблон:Math, then Шаблон:Math is nonnegative; if Шаблон:Math is invertible, then Шаблон:Math is positive. A converse is also true in the sense that, for a non-negative operator Шаблон:Math, there exists a unique non-negative square root Шаблон:Math such that <math display="block">A = B^2 = B^*B\,.</math>

In a sense made precise by the spectral theorem, self-adjoint operators can usefully be thought of as operators that are "real". An element Шаблон:Math of Шаблон:Math is called normal if Шаблон:Math. Normal operators decompose into the sum of a self-adjoint operator and an imaginary multiple of a self adjoint operator <math display="block">A = \frac{A + A^*}{2} + i\frac{A - A^*}{2i}</math> that commute with each other. Normal operators can also usefully be thought of in terms of their real and imaginary parts.

An element Шаблон:Math of Шаблон:Math is called unitary if Шаблон:Math is invertible and its inverse is given by Шаблон:Math. This can also be expressed by requiring that Шаблон:Math be onto and Шаблон:Math for all Шаблон:Math. The unitary operators form a group under composition, which is the isometry group of Шаблон:Math.

An element of Шаблон:Math is compact if it sends bounded sets to relatively compact sets. Equivalently, a bounded operator Шаблон:Math is compact if, for any bounded sequence Шаблон:Math, the sequence Шаблон:Math has a convergent subsequence. Many integral operators are compact, and in fact define a special class of operators known as Hilbert–Schmidt operators that are especially important in the study of integral equations. Fredholm operators differ from a compact operator by a multiple of the identity, and are equivalently characterized as operators with a finite dimensional kernel and cokernel. The index of a Fredholm operator Шаблон:Math is defined by <math display="block">\operatorname{index} T = \dim\ker T - \dim\operatorname{coker} T \,.</math>

The index is homotopy invariant, and plays a deep role in differential geometry via the Atiyah–Singer index theorem.

Unbounded operators

Unbounded operators are also tractable in Hilbert spaces, and have important applications to quantum mechanics. An unbounded operator Шаблон:Math on a Hilbert space Шаблон:Math is defined as a linear operator whose domain Шаблон:Math is a linear subspace of Шаблон:Math. Often the domain Шаблон:Math is a dense subspace of Шаблон:Math, in which case Шаблон:Math is known as a densely defined operator.

The adjoint of a densely defined unbounded operator is defined in essentially the same manner as for bounded operators. Self-adjoint unbounded operators play the role of the observables in the mathematical formulation of quantum mechanics. Examples of self-adjoint unbounded operators on the Hilbert space Шаблон:Math are:

• A suitable extension of the differential operator <math display="block">(A f)(x) = -i \frac{\mathrm{d}}{\mathrm{d}x} f(x) \,,</math> where Шаблон:Math is the imaginary unit and Шаблон:Math is a differentiable function of compact support.
• The multiplication-by-Шаблон:Math operator: <math display="block">(B f) (x) = x f(x)\,. </math>

These correspond to the momentum and position observables, respectively. Neither Шаблон:Math nor Шаблон:Math is defined on all of Шаблон:Math, since in the case of Шаблон:Math the derivative need not exist, and in the case of Шаблон:Math the product function need not be square integrable. In both cases, the set of possible arguments form dense subspaces of Шаблон:Math.

Constructions

Direct sums

Two Hilbert spaces Шаблон:Math and Шаблон:Math can be combined into another Hilbert space, called the (orthogonal) direct sum, and denoted <math display="block">H_1 \oplus H_2 \,,</math>

consisting of the set of all ordered pairs Шаблон:Math where Шаблон:Math, Шаблон:Math, and inner product defined by <math display="block">\bigl\langle (x_1, x_2), (y_1, y_2)\bigr\rangle_{H_1 \oplus H_2} = \left\langle x_1, y_1\right\rangle_{H_1} + \left\langle x_2, y_2\right\rangle_{H_2} \,.</math>

More generally, if Шаблон:Math is a family of Hilbert spaces indexed by Шаблон:Nowrap, then the direct sum of the Шаблон:Math, denoted <math display="block">\bigoplus_{i \in I}H_i</math> consists of the set of all indexed families <math display="block">x = (x_i \in H_i \mid i \in I) \in \prod_{i \in I}H_i</math> in the Cartesian product of the Шаблон:Math such that <math display="block">\sum_{i \in I} \|x_i\|^2 < \infty \,.</math>

The inner product is defined by <math display="block">\langle x, y\rangle = \sum_{i \in I} \left\langle x_i, y_i\right\rangle_{H_i} \,.</math>

Each of the Шаблон:Math is included as a closed subspace in the direct sum of all of the Шаблон:Math. Moreover, the Шаблон:Math are pairwise orthogonal. Conversely, if there is a system of closed subspaces, Шаблон:Math, Шаблон:Math, in a Hilbert space Шаблон:Math, that are pairwise orthogonal and whose union is dense in Шаблон:Math, then Шаблон:Math is canonically isomorphic to the direct sum of Шаблон:Math. In this case, Шаблон:Math is called the internal direct sum of the Шаблон:Math. A direct sum (internal or external) is also equipped with a family of orthogonal projections Шаблон:Math onto the Шаблон:Mathth direct summand Шаблон:Math. These projections are bounded, self-adjoint, idempotent operators that satisfy the orthogonality condition <math display="block">E_i E_j = 0,\quad i \neq j \,.</math>

The spectral theorem for compact self-adjoint operators on a Hilbert space Шаблон:Math states that Шаблон:Math splits into an orthogonal direct sum of the eigenspaces of an operator, and also gives an explicit decomposition of the operator as a sum of projections onto the eigenspaces. The direct sum of Hilbert spaces also appears in quantum mechanics as the Fock space of a system containing a variable number of particles, where each Hilbert space in the direct sum corresponds to an additional degree of freedom for the quantum mechanical system. In representation theory, the Peter–Weyl theorem guarantees that any unitary representation of a compact group on a Hilbert space splits as the direct sum of finite-dimensional representations.

Tensor products

Шаблон:Main If Шаблон:Math and Шаблон:Math, then one defines an inner product on the (ordinary) tensor product as follows. On simple tensors, let <math display="block"> \langle x_1 \otimes x_2, \, y_1 \otimes y_2 \rangle = \langle x_1, y_1 \rangle \, \langle x_2, y_2 \rangle \,.</math>

This formula then extends by sesquilinearity to an inner product on Шаблон:Math. The Hilbertian tensor product of Шаблон:Math and Шаблон:Math, sometimes denoted by Шаблон:Math, is the Hilbert space obtained by completing Шаблон:Math for the metric associated to this inner product.

An example is provided by the Hilbert space Шаблон:Math. The Hilbertian tensor product of two copies of Шаблон:Math is isometrically and linearly isomorphic to the space Шаблон:Math of square-integrable functions on the square Шаблон:Math. This isomorphism sends a simple tensor Шаблон:Math to the function <math display="block">(s, t) \mapsto f_1(s) \, f_2(t)</math> on the square.

This example is typical in the following sense. Associated to every simple tensor product Шаблон:Math is the rank one operator from Шаблон:Math to Шаблон:Math that maps a given Шаблон:Math as <math display="block">x^* \mapsto x^*(x_1) x_2 \,.</math>

This mapping defined on simple tensors extends to a linear identification between Шаблон:Math and the space of finite rank operators from Шаблон:Math to Шаблон:Math. This extends to a linear isometry of the Hilbertian tensor product Шаблон:Math with the Hilbert space Шаблон:Math of Hilbert–Schmidt operators from Шаблон:Math to Шаблон:Math.

Orthonormal bases

The notion of an orthonormal basis from linear algebra generalizes over to the case of Hilbert spaces. In a Hilbert space Шаблон:Math, an orthonormal basis is a family Шаблон:Math of elements of Шаблон:Math satisfying the conditions:

1. Orthogonality: Every two different elements of Шаблон:Math are orthogonal: Шаблон:Math for all Шаблон:Math with Шаблон:Nowrap.
2. Normalization: Every element of the family has norm 1: Шаблон:Math for all Шаблон:Math.
3. Completeness: The linear span of the family Шаблон:Math, Шаблон:Math, is dense in H.

A system of vectors satisfying the first two conditions basis is called an orthonormal system or an orthonormal set (or an orthonormal sequence if Шаблон:Math is countable). Such a system is always linearly independent.

Despite the name, an orthonormal basis is not, in general, a basis in the sense of linear algebra (Hamel basis). More precisely, an orthonormal basis is a Hamel basis if and only if the Hilbert space is a finite-dimensional vector space.

Completeness of an orthonormal system of vectors of a Hilbert space can be equivalently restated as:

for every Шаблон:Math, if Шаблон:Math for all Шаблон:Math, then Шаблон:Math.

This is related to the fact that the only vector orthogonal to a dense linear subspace is the zero vector, for if Шаблон:Math is any orthonormal set and Шаблон:Math is orthogonal to Шаблон:Math, then Шаблон:Math is orthogonal to the closure of the linear span of Шаблон:Math, which is the whole space.

Examples of orthonormal bases include:

• the set Шаблон:Math forms an orthonormal basis of Шаблон:Math with the dot product;
• the sequence Шаблон:Math with Шаблон:Math forms an orthonormal basis of the complex space Шаблон:Math;

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 implies that every vector in the space can be written as the sum of an infinite series, and the orthogonality implies that this decomposition is unique.

Sequence spaces

The space <math>\ell_2</math> of square-summable sequences of complex numbers is the set of infinite sequences <math display="block">(c_1, c_2, c_3, \dots)</math> of real or complex numbers such that <math display="block">\left|c_1\right|^2 + \left|c_2\right|^2 + \left|c_3\right|^2 + \cdots < \infty \,.</math>

This space has an orthonormal basis: <math display="block">\begin{align}

 e_1 &= (1, 0, 0, \dots) \\
 e_2 &= (0, 1, 0, \dots) \\
     & \ \ \vdots

\end{align}</math>

This space is the infinite-dimensional generalization of the <math>\ell_2^n</math> space of finite-dimensional vectors. It is usually the first example used to show that in infinite-dimensional spaces, a set that is closed and bounded is not necessarily (sequentially) compact (as is the case in all finite dimensional spaces). Indeed, the set of orthonormal vectors above shows this: It is an infinite sequence of vectors in the unit ball (i.e., the ball of points with norm less than or equal one). This set is clearly bounded and closed; yet, no subsequence of these vectors converges to anything and consequently the unit ball in <math>\ell_2</math> is not compact. Intuitively, this is because "there is always another coordinate direction" into which the next elements of the sequence can evade.

One can generalize the space <math>\ell_2</math> in many ways. For example, if Шаблон:Math is any set, then one can form a Hilbert space of sequences with index set Шаблон:Math, defined by <math display="block">\ell^2(B) =\biggl\{ x : B \xrightarrow{x} \mathbb{C} \mathrel{\bigg|} \sum_{b \in B} \left|x (b)\right|^2 < \infty \biggr\} \,.</math>

The summation over B is here defined by <math display="block">\sum_{b \in B} \left|x (b)\right|^2 = \sup \sum_{n=1}^N \left|x(b_n)\right|^2</math> the supremum being taken over all finite subsets of Шаблон:Math. It follows that, for this sum to be finite, every element of Шаблон:Math has only countably many nonzero terms. This space becomes a Hilbert space with the inner product <math display="block">\langle x, y \rangle = \sum_{b \in B} x(b)\overline{y(b)}</math>

for all Шаблон:Math. Here the sum also has only countably many nonzero terms, and is unconditionally convergent by the Cauchy–Schwarz inequality.

An orthonormal basis of Шаблон:Math is indexed by the set Шаблон:Math, given by

<math display="block">e_b(b') = \begin{cases}

 1 & \text{if } b=b'\\
 0 & \text{otherwise.}

\end{cases}</math>

Bessel's inequality and Parseval's formula

Let Шаблон:Math be a finite orthonormal system in Шаблон:Math. For an arbitrary vector Шаблон:Math, let <math display="block">y = \sum_{j=1}^n \langle x, f_j \rangle \, f_j \,.</math>

Then Шаблон:Math for every Шаблон:Math. It follows that Шаблон:Math is orthogonal to each Шаблон:Math, hence Шаблон:Math is orthogonal to Шаблон:Math. Using the Pythagorean identity twice, it follows that <math display="block">\|x\|^2 = \|x - y\|^2 + \|y\|^2 \ge \|y\|^2 = \sum_{j=1}^n\bigl|\langle x, f_j \rangle\bigr|^2 \,.</math>

Let Шаблон:Math, be an arbitrary orthonormal system in Шаблон:Math. Applying the preceding inequality to every finite subset Шаблон:Math of Шаблон:Math gives Bessel's inequality: <math display="block">\sum_{i \in I}\bigl|\langle x, f_i \rangle\bigr|^2 \le \|x\|^2, \quad x \in H</math> (according to the definition of the sum of an arbitrary family of non-negative real numbers).

Geometrically, Bessel's inequality implies that the orthogonal projection of Шаблон:Math onto the linear subspace spanned by the Шаблон:Math has norm that does not exceed that of Шаблон:Math. In two dimensions, this is the assertion that the length of the leg of a right triangle may not exceed the length of the hypotenuse.

Bessel's inequality is a stepping stone to the stronger result called Parseval's identity, which governs the case when Bessel's inequality is actually an equality. By definition, if Шаблон:Math is an orthonormal basis of Шаблон:Math, then every element Шаблон:Math of Шаблон:Math may be written as <math display="block">x = \sum_{k \in B} \left\langle x, e_k \right\rangle \, e_k \,.</math>

Even if Шаблон:Math is uncountable, Bessel's inequality guarantees that the expression is well-defined and consists only of countably many nonzero terms. This sum is called the Fourier expansion of Шаблон:Math, and the individual coefficients Шаблон:Math are the Fourier coefficients of Шаблон:Math. Parseval's identity then asserts that <math display="block">\|x\|^2 = \sum_{k\in B}|\langle x, e_k\rangle|^2 \,.</math>

Conversely, if Шаблон:Math is an orthonormal set such that Parseval's identity holds for every Шаблон:Math, then Шаблон:Math is an orthonormal basis.

Hilbert dimension

As a consequence of Zorn's lemma, 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. For instance, since Шаблон:Math has an orthonormal basis indexed by Шаблон:Math, its Hilbert dimension is the cardinality of Шаблон:Math (which may be a finite integer, or a countable or uncountable cardinal number).

The Hilbert dimension is not greater than the Hamel dimension (the usual dimension of a vector space). The two dimensions are equal if and only one of them is finite.

As a consequence of Parseval's identity, if Шаблон:Math is an orthonormal basis of Шаблон:Math, then the map Шаблон:Math defined by Шаблон:Math is an isometric isomorphism of Hilbert spaces: it is a bijective linear mapping such that <math display="block">\bigl\langle \Phi (x), \Phi(y) \bigr\rangle_{l^2(B)} = \left\langle x, y \right\rangle_H</math> for all Шаблон:Math. The cardinal number of Шаблон:Math is the Hilbert dimension of Шаблон:Math. Thus every Hilbert space is isometrically isomorphic to a sequence space Шаблон:Math for some set Шаблон:Math.

Separable spaces

By definition, a Hilbert space is separable provided it contains a dense countable subset. Along with Zorn's lemma, this means a Hilbert space is separable if and only if it admits a countable orthonormal basis. All infinite-dimensional separable Hilbert spaces are therefore isometrically isomorphic to the square-summable sequence space <math>\ell^2.</math>

In the past, Hilbert spaces were often required to be separable as part of the definition.

In quantum field theory

Most spaces used in physics are separable, and since these are all isomorphic to each other, one often refers to any infinite-dimensional separable Hilbert space as "the Hilbert space" or just "Hilbert space". Even in quantum field theory, most of the Hilbert spaces are in fact separable, as stipulated by the Wightman axioms. However, it is sometimes argued that non-separable Hilbert spaces are also important in quantum field theory, roughly because the systems in the theory possess an infinite number of degrees of freedom and any infinite Hilbert tensor product (of spaces of dimension greater than one) is non-separable. For instance, a bosonic field can be naturally thought of as an element of a tensor product whose factors represent harmonic oscillators at each point of space. From this perspective, the natural state space of a boson might seem to be a non-separable space. However, it is only a small separable subspace of the full tensor product that can contain physically meaningful fields (on which the observables can be defined). Another non-separable Hilbert space models the state of an infinite collection of particles in an unbounded region of space. An orthonormal basis of the space is indexed by the density of the particles, a continuous parameter, and since the set of possible densities is uncountable, the basis is not countable.

Orthogonal complements and projections

If Шаблон:Math is a subset of a Hilbert space Шаблон:Math, the set of vectors orthogonal to Шаблон:Math is defined by <math display="block">S^\perp = \left\{ x \in H \mid \langle x, s \rangle = 0\ \text{ for all } s \in S \right\} \,.</math>

The set Шаблон:Math is a closed subspace of Шаблон:Math (can be proved easily using the linearity and continuity of the inner product) and so forms itself a Hilbert space. If Шаблон:Math is a closed subspace of Шаблон:Math, then Шаблон:Math is called the Шаблон:Em of Шаблон:Math. In fact, every Шаблон:Math can then be written uniquely as Шаблон:Math, with Шаблон:Math and Шаблон:Math. Therefore, Шаблон:Math is the internal Hilbert direct sum of Шаблон:Math and Шаблон:Math.

The linear operator Шаблон:Math that maps Шаблон:Math to Шаблон:Math is called the Шаблон:Em onto Шаблон:Math. There is a natural one-to-one correspondence between the set of all closed subspaces of Шаблон:Math and the set of all bounded self-adjoint operators Шаблон:Math such that Шаблон:Math. Specifically,

This provides the geometrical interpretation of Шаблон:Math: it is the best approximation to x by elements of V.

Projections Шаблон:Math and Шаблон:Math are called mutually orthogonal if Шаблон:Math. This is equivalent to Шаблон:Math and Шаблон:Math being orthogonal as subspaces of Шаблон:Math. The sum of the two projections Шаблон:Math and Шаблон:Math is a projection only if Шаблон:Math and Шаблон:Math are orthogonal to each other, and in that case Шаблон:Math. The composite Шаблон:Math is generally not a projection; in fact, the composite is a projection if and only if the two projections commute, and in that case Шаблон:Math.

By restricting the codomain to the Hilbert space Шаблон:Math, the orthogonal projection Шаблон:Math gives rise to a projection mapping Шаблон:Math; it is the adjoint of the inclusion mapping <math display="block">i : V \to H \,,</math> meaning that <math display="block">\left\langle i x, y\right\rangle_H = \left\langle x, \pi y \right\rangle_V</math> for all Шаблон:Math and Шаблон:Math.

The operator norm of the orthogonal projection Шаблон:Math onto a nonzero closed subspace Шаблон:Math is equal to 1: <math display="block">\|P_V\| = \sup_{x \in H, x \neq 0} \frac{\|P_V x\|}{\|x\|} = 1 \,.</math>

Every closed subspace V of a Hilbert space is therefore the image of an operator Шаблон:Math of norm one such that Шаблон:Math. The property of possessing appropriate projection operators characterizes Hilbert spaces:

• A Banach space of dimension higher than 2 is (isometrically) a Hilbert space if and only if, for every closed subspace Шаблон:Math, there is an operator Шаблон:Math of norm one whose image is Шаблон:Math such that Шаблон:Math.

While this result characterizes the metric structure of a Hilbert space, the structure of a Hilbert space as a topological vector space can itself be characterized in terms of the presence of complementary subspaces:

• A Banach space Шаблон:Math is topologically and linearly isomorphic to a Hilbert space if and only if, to every closed subspace Шаблон:Math, there is a closed subspace Шаблон:Math such that Шаблон:Math is equal to the internal direct sum Шаблон:Math.

The orthogonal complement satisfies some more elementary results. It is a monotone function in the sense that if Шаблон:Math, then Шаблон:Math with equality holding if and only if Шаблон:Math is contained in the closure of Шаблон:Math. This result is a special case of the Hahn–Banach theorem. The closure of a subspace can be completely characterized in terms of the orthogonal complement: if Шаблон:Math is a subspace of Шаблон:Math, then the closure of Шаблон:Math is equal to Шаблон:Math. The orthogonal complement is thus a Galois connection on the partial order of subspaces of a Hilbert space. In general, the orthogonal complement of a sum of subspaces is the intersection of the orthogonal complements: <math display="block">\biggl(\sum_i V_i\biggr)^\perp = \bigcap_i V_i^\perp \,.</math>

If the Шаблон:Math are in addition closed, then <math display="block">\overline{\sum_i V_i^\perp \vphantom\Big|} = \biggl(\bigcap_i V_i\biggr)^\perp \,.</math>

Spectral theory

There is a well-developed spectral theory for self-adjoint operators in a Hilbert space, that is roughly analogous to the study of symmetric matrices over the reals or self-adjoint matrices over the complex numbers. In the same sense, one can obtain a "diagonalization" of a self-adjoint operator as a suitable sum (actually an integral) of orthogonal projection operators.

The spectrum of an operator Шаблон:Math, denoted Шаблон:Math, is the set of complex numbers Шаблон:Math such that Шаблон:Math lacks a continuous inverse. If Шаблон:Math is bounded, then the spectrum is always a compact set in the complex plane, and lies inside the disc Шаблон:Math. If Шаблон:Math is self-adjoint, then the spectrum is real. In fact, it is contained in the interval Шаблон:Math where <math display="block">m = \inf_{\|x\|=1}\langle Tx, x\rangle \,,\quad M = \sup_{\|x\|=1}\langle Tx, x\rangle \,.</math>

Moreover, Шаблон:Math and Шаблон:Math are both actually contained within the spectrum.

The eigenspaces of an operator Шаблон:Math are given by <math display="block">H_\lambda = \ker(T - \lambda)\,.</math>

Unlike with finite matrices, not every element of the spectrum of Шаблон:Math must be an eigenvalue: the linear operator Шаблон:Math may only lack an inverse because it is not surjective. Elements of the spectrum of an operator in the general sense are known as spectral values. Since spectral values need not be eigenvalues, the spectral decomposition is often more subtle than in finite dimensions.

However, the spectral theorem of a self-adjoint operator Шаблон:Math takes a particularly simple form if, in addition, Шаблон:Math is assumed to be a compact operator. The spectral theorem for compact self-adjoint operators states:

• A compact self-adjoint operator Шаблон:Math has only countably (or finitely) many spectral values. The spectrum of Шаблон:Math has no limit point in the complex plane except possibly zero. The eigenspaces of Шаблон:Math decompose Шаблон:Math into an orthogonal direct sum: <math display="block">H = \bigoplus_{\lambda\in\sigma(T)}H_\lambda \,.</math> Moreover, if Шаблон:Math denotes the orthogonal projection onto the eigenspace Шаблон:Math, then <math display="block">T = \sum_{\lambda\in\sigma(T)} \lambda E_\lambda \,,</math> where the sum converges with respect to the norm on Шаблон:Math.

This theorem plays a fundamental role in the theory of integral equations, as many integral operators are compact, in particular those that arise from Hilbert–Schmidt operators.

The general spectral theorem for self-adjoint operators involves a kind of operator-valued Riemann–Stieltjes integral, rather than an infinite summation. The spectral family associated to Шаблон:Math associates to each real number λ an operator Шаблон:Math, which is the projection onto the nullspace of the operator Шаблон:Math, where the positive part of a self-adjoint operator is defined by <math display="block">A^+ = \tfrac{1}{2}\Bigl(\sqrt{A^2} + A\Bigr) \,.</math>

The operators Шаблон:Math are monotone increasing relative to the partial order defined on self-adjoint operators; the eigenvalues correspond precisely to the jump discontinuities. One has the spectral theorem, which asserts <math display="block">T = \int_\mathbb{R} \lambda\, \mathrm{d}E_\lambda \,.</math>

The integral is understood as a Riemann–Stieltjes integral, convergent with respect to the norm on Шаблон:Math. In particular, one has the ordinary scalar-valued integral representation <math display="block">\langle Tx, y\rangle = \int_{\R} \lambda\,\mathrm{d}\langle E_\lambda x, y\rangle \,.</math>

A somewhat similar spectral decomposition holds for normal operators, although because the spectrum may now contain non-real complex numbers, the operator-valued Stieltjes measure Шаблон:Math must instead be replaced by a resolution of the identity.

A major application of spectral methods is the spectral mapping theorem, which allows one to apply to a self-adjoint operator Шаблон:Math any continuous complex function Шаблон:Math defined on the spectrum of Шаблон:Math by forming the integral <math display="block">f(T) = \int_{\sigma(T)} f(\lambda)\,\mathrm{d}E_\lambda \,.</math>

The resulting continuous functional calculus has applications in particular to pseudodifferential operators.

The spectral theory of unbounded self-adjoint operators is only marginally more difficult than for bounded operators. The spectrum of an unbounded operator is defined in precisely the same way as for bounded operators: Шаблон:Math is a spectral value if the resolvent operator <math display="block">R_\lambda = (T - \lambda)^{-1}</math>

fails to be a well-defined continuous operator. The self-adjointness of Шаблон:Math still guarantees that the spectrum is real. Thus the essential idea of working with unbounded operators is to look instead at the resolvent Шаблон:Math where Шаблон:Math is nonreal. This is a bounded normal operator, which admits a spectral representation that can then be transferred to a spectral representation of Шаблон:Math itself. A similar strategy is used, for instance, to study the spectrum of the Laplace operator: rather than address the operator directly, one instead looks as an associated resolvent such as a Riesz potential or Bessel potential.

A precise version of the spectral theorem in this case is: Шаблон:Math theorem There is also a version of the spectral theorem that applies to unbounded normal operators.

In popular culture

In Gravity's Rainbow (1973), a novel by Thomas Pynchon, one of the characters is called "Sammy Hilbert-Spaess", a pun on "Hilbert Space". The novel refers also to Gödel's incompleteness theorems.

See also

• Шаблон:Annotated link
• Шаблон:Annotated link
• Шаблон:Annotated link
• Шаблон:Annotated link
• Шаблон:Annotated link
• Шаблон:Annotated link
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• Шаблон:Annotated link
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• Шаблон:Annotated link
• Шаблон:Annotated link
• Шаблон:Annotated link
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• Шаблон:Annotated link

References

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External links

• Шаблон:Springer
• Hilbert space at Mathworld
• 245B, notes 5: Hilbert spaces by Terence Tao