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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Hilbert spaces (named after
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...
) allow generalizing the methods of
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrice ...
and
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
from (finite-dimensional)
Euclidean vector space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean s ...
s to spaces that may be
infinite-dimensional In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to d ...
. Hilbert spaces arise naturally and frequently in mathematics and
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
, typically as
function space In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vect ...
s. Formally, a Hilbert space is a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
equipped with an
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
that defines a distance function for which the space is a
complete metric space In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the bou ...
. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...
, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to h ...
s,
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
,
Fourier analysis In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph ...
(which includes applications to
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
and heat transfer), and
ergodic theory Ergodic theory ( Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expr ...
(which forms the mathematical underpinning of
thermodynamics Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws ...
).
John von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest c ...
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 Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
. Apart from the classical Euclidean vector spaces, examples of Hilbert spaces include spaces of square-integrable functions, spaces of sequences,
Sobolev space In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...
s consisting of
generalized function In mathematics, generalized functions are objects extending the notion of functions. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful in making discontinuous functions ...
s, and
Hardy space In complex analysis, the Hardy spaces (or Hardy classes) ''Hp'' are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper . I ...
s of
holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex deriv ...
s. Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the
Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposit ...
and parallelogram law hold in a Hilbert space. At a deeper level, perpendicular projection onto a linear subspace or a subspace (the analog of " dropping the altitude" of a triangle) 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 mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For examp ...
, in analogy with
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
in classical geometry. When this basis is
countably infinite In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural number ...
, it allows identifying the Hilbert space with the space of the
infinite sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called t ...
s 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 Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean s ...
consisting of three-dimensional vectors, denoted by , and equipped with the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alg ...
. The dot product takes two vectors and , and produces a real number . If and are represented in
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
, then the dot product is defined by \begin x_1 \\ x_2 \\ x_3 \end \cdot \begin y_1 \\ y_2 \\ y_3 \end = x_1 y_1 + x_2 y_2 + x_3 y_3 \,. The dot product satisfies the properties: # It is symmetric in and : . # It is
linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
in its first argument: for any scalars , , and vectors , , and . # It is
positive definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular: * Positive-definite bilinear form * Positive-definite fu ...
: for all vectors , , with equality
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
. An operation on pairs of vectors that, like the dot product, satisfies these three properties is known as a (real)
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
. A
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
equipped with such an inner product is known as a (real)
inner product space In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
. 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 , and to the angle between two vectors and by means of the formula \mathbf\cdot\mathbf = \left\, \mathbf\right\, \left\, \mathbf\right\, \, \cos\theta \,.
Multivariable calculus Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one Variable (mathematics), variable to calculus with Function of several real variables, functions of several variables: the Differential calculus, di ...
in Euclidean space relies on the ability to compute limits, and to have useful criteria for concluding that limits exist. A
mathematical series In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
\sum_^\infty \mathbf_n consisting of vectors in is
absolutely convergent In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is s ...
provided that the sum of the lengths converges as an ordinary series of real numbers: \sum_^\infty \, \mathbf_k\, < \infty \,. 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 \left\, \mathbf - \sum_^N \mathbf_k \right\, \to 0 \quad \text N \to\infty \,. 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 number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s. The
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
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 In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
: , z, ^2 = z\overline \,. If is a decomposition of into its real and imaginary parts, then the modulus is the usual Euclidean two-dimensional length: , z, = \sqrt \,. The inner product of a pair of complex numbers and is the product of with the complex conjugate of : \langle z, w\rangle = z\overline\,. This is complex-valued. The real part of gives the usual two-dimensional Euclidean
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alg ...
. A second example is the space whose elements are pairs of complex numbers . Then the inner product of with another such vector is given by \langle z, w\rangle = z_1\overline + z_2\overline\,. The real part of is then the two-dimensional Euclidean dot product. This inner product is ''Hermitian'' symmetric, which means that the result of interchanging and is the complex conjugate: \langle w, z\rangle = \overline\,.


Definition

A is a
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
or complex
inner product space In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
that is also a
complete metric space In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the bou ...
with respect to the distance function induced by the inner product.The mathematical material in this section can be found in any good textbook on functional analysis, such as , , or . To say that is a means that is a complex vector space on which there is an inner product \langle x, y \rangle associating a complex number to each pair of elements x, y 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 In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
of the inner product of the swapped elements: \langle y, x\rangle = \overline\,. Importantly, this implies that \langle x, x\rangle is a real number. # The inner product is
linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
in its firstIn some conventions, inner products are linear in their second arguments instead. argument. For all complex numbers a and b, \langle ax_1 + bx_2, y\rangle = a\langle x_1, y\rangle + b\langle x_2, y\rangle\,. # The inner product of an element with itself is
positive definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular: * Positive-definite bilinear form * Positive-definite fu ...
: \begin \langle x, x\rangle > 0 & \quad \text x \neq 0, \\ \langle x, x\rangle = 0 & \quad \text x = 0\,. \end It follows from properties 1 and 2 that a complex inner product is , also called , in its second argument, meaning that \langle x, ay_1 + by_2\rangle = \bar\langle x, y_1\rangle + \bar\langle x, y_2\rangle\,. 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 In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example. Definition Vector spaces Let V, W ...
and (H, H, \langle \cdot, \cdot \rangle) will form a dual system. The norm is the real-valued function \, x\, = \sqrt\,, and the distance d between two points x, y in is defined in terms of the norm by d(x, y) = \, x - y\, = \sqrt\,. That this function is a distance function means firstly that it is symmetric in x and y, secondly that the distance between x and itself is zero, and otherwise the distance between x and y must be positive, and lastly that the
triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but ...
holds, meaning that the length of one leg of a triangle cannot exceed the sum of the lengths of the other two legs: d(x, z) \leq d(x, y) + d(y, z)\,. : This last property is ultimately a consequence of the more fundamental
Cauchy–Schwarz inequality The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics. The inequality for sums was published by . The corresponding inequality f ...
, which asserts \left, \langle x, y\rangle\ \leq \, x\, \, y\, with equality if and only if x and y are
linearly dependent In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
. With a distance function defined in this way, any inner product space is a
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
, and sometimes is known as a . Any pre-Hilbert space that is additionally also a
complete space In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the bou ...
is a Hilbert space. The of is expressed using a form of the
Cauchy criterion The Cauchy convergence test is a method used to test infinite series for convergence. It relies on bounding sums of terms in the series. This convergence criterion is named after Augustin-Louis Cauchy who published it in his textbook Cours d'Analy ...
for sequences in : a pre-Hilbert space is complete if every
Cauchy sequence In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numbe ...
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 \sum_^\infty u_k converges absolutely in the sense that \sum_^\infty\, u_k\, < \infty\,, 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 space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
s. As such they are
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
s, in which topological notions like the
openness Openness is an overarching concept or philosophy that is characterized by an emphasis on transparency and collaboration. That is, openness refers to "accessibility of knowledge, technology and other resources; the transparency of action; the per ...
and closedness of subsets are well defined. Of special importance is the notion of a closed
linear subspace In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, l ...
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 In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural nu ...
consists of all infinite sequences of complex numbers such that the following series converges: \sum_^\infty , z_n, ^2 The inner product on is defined by: \langle \mathbf, \mathbf\rangle = \sum_^\infty z_n\overline\,, This second series converges as a consequence of the
Cauchy–Schwarz inequality The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics. The inequality for sums was published by . The corresponding inequality f ...
and the convergence of the previous series. Completeness of the space holds provided that whenever a series of elements from converges absolutely (in norm), then it converges to an element of . The proof is basic in
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied ...
, 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

Prior to the development of Hilbert spaces, other generalizations of Euclidean spaces were known to
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...
s and
physicist A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe. Physicists generally are interested in the root or ultimate cau ...
s. 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 Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
or
complex numbers In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
) 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 David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...
and Erhard Schmidt's study of integral equations, that two
square-integrable In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value ...
real-valued functions and on an interval have an ''inner product'' : \langle f, g \rangle = \int_a^b f(x)g(x)\, \mathrmx which has many of the familiar properties of the Euclidean dot product. In particular, the idea of an
orthogonal In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
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 : f(x) \mapsto \int_a^b K(x, y) f(y)\, \mathrmy where is a continuous function symmetric in and . The resulting
eigenfunction expansion In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
expresses the function as a series of the form : K(x, y) = \sum_n \lambda_n\varphi_n(x)\varphi_n(y) 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 In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Lebe ...
, an alternative to the
Riemann integral In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of G� ...
introduced by
Henri Lebesgue Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of ...
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 In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the bou ...
. As a consequence of the interplay between geometry and completeness, the 19th century results of
Joseph Fourier Jean-Baptiste Joseph Fourier (; ; 21 March 1768 – 16 May 1830) was a French mathematician and physicist born in Auxerre and best known for initiating the investigation of Fourier series, which eventually developed into Fourier analysis and ha ...
, Friedrich Bessel and
Marc-Antoine Parseval Marc-Antoine Parseval des Chênes (27 April 1755 – 16 August 1836) was a French mathematician, most famous for what is now known as Parseval's theorem, which presaged the unitarity of the Fourier transform. He was born in Rosières-aux-Sali ...
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 :''This article describes a theorem concerning the dual of a Hilbert space. For the theorems relating linear functionals to Measure (mathematics), measures, see Riesz–Markov–Kakutani representation theorem.'' The Riesz representation theorem, ...
was independently established by Maurice Fréchet and Frigyes Riesz in 1907.
John von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest c ...
coined the term ''abstract Hilbert space'' in his work on unbounded
Hermitian operators In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to its ...
. Although other mathematicians such as
Hermann Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is asso ...
and
Norbert Wiener Norbert Wiener (November 26, 1894 – March 18, 1964) was an American mathematician and philosopher. He was a professor of mathematics at the Massachusetts Institute of Technology (MIT). A child prodigy, Wiener later became an early researcher ...
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 Eugene Paul "E. P." Wigner ( hu, Wigner Jenő Pál, ; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist who also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his co ...
. 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 operator In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to i ...
s on that space, the symmetries of the system are
unitary operator In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Unitary operators are usually taken as operating ''on'' a Hilbert space, but the same notion serves to define the co ...
s, and
measurements Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events. In other words, measurement is a process of determining how large or small a physical quantity is as compared ...
are
orthogonal projection In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it wer ...
s. The relation between quantum mechanical symmetries and unitary operators provided an impetus for the development of the
unitary Unitary may refer to: Mathematics * Unitary divisor * Unitary element * Unitary group * Unitary matrix * Unitary morphism * Unitary operator * Unitary transformation * Unitary representation In mathematics, a unitary representation of a grou ...
representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
of
groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
, 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 In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a ...
can be analyzed using Hilbert space techniques in the framework of
ergodic theory Ergodic theory ( Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expr ...
. The algebra of
observable In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum phy ...
s in quantum mechanics is naturally an algebra of operators defined on a Hilbert space, according to
Werner Heisenberg Werner Karl Heisenberg () (5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the main pioneers of the theory of quantum mechanics. He published his work in 1925 in a Über quantentheoretische Umdeutung kinematis ...
's
matrix mechanics Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. It was the first conceptually autonomous and logically consistent formulation of quantum mechanics. Its account of quantum j ...
formulation of quantum theory. Von Neumann began investigating
operator algebra In functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with the multiplication given by the composition of mappings. The results obtained in the study of ...
s 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 algebra In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra. Von Neumann algebra ...
s. In the 1940s, Israel Gelfand, Mark Naimark and
Irving Segal Irving Ezra Segal (1918–1998) was an American mathematician known for work on theoretical quantum mechanics. He shares credit for what is often referred to as the Segal–Shale–Weil representation. Early in his career Segal became known for h ...
gave a definition of a kind of operator algebras called
C*-algebra In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuou ...
s 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 space In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vect ...
s associated to measure spaces , where is a set, is a
σ-algebra In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set ''X'' is a collection Σ of subsets of ''X'' that includes the empty subset, is closed under complement, and is closed under countable unions and countabl ...
of subsets of , and is a
countably additive measure In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many s ...
on . Let be the space of those complex-valued measurable functions on for which the
Lebesgue integral In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Lebe ...
of the square of the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
of the function is finite, i.e., for a function in , \int_X , f, ^2 \mathrm \mu < \infty \,, and where functions are identified if and only if they differ only on a set of measure zero. The inner product of functions and in is then defined as \langle f, g\rangle = \int_X f(t) \overline \, \mathrm \mu(t) or \langle f, g\rangle = \int_X \overline g(t) \, \mathrm \mu(t) \,, where the second form (conjugation of the first element) is commonly found in the theoretical physics literature. For and in , the integral exists because of the Cauchy–Schwarz inequality, and defines an inner product on the space. Equipped with this inner product, 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 and of square-integrable functions with respect to the
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wi ...
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 is any positive measurable function, the space of all measurable functions on the interval satisfying \int_0^1 \bigl, f(t)\bigr, ^2 w(t)\, \mathrmt < \infty is called the weighted space , and is called the weight function. The inner product is defined by \langle f, g\rangle = \int_0^1 f(t) \overline w(t) \, \mathrmt \,. The weighted space is identical with the Hilbert space where the measure of a Lebesgue-measurable set is defined by \mu(A) = \int_A w(t)\,\mathrmt \,. Weighted 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 space In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...
s, denoted by or , are Hilbert spaces. These are a special kind of
function space In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vect ...
in which differentiation may be performed, but that (unlike other
Banach spaces In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
such as the
Hölder space Hölder: * ''Hölder, Hoelder'' as surname * Hölder condition * Hölder's inequality * Hölder mean * Jordan–Hölder theorem In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a mo ...
s) support the structure of an inner product. Because differentiation is permitted, Sobolev spaces are a convenient setting for the theory of
partial differential equations In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...
. They also form the basis of the theory of
direct methods in the calculus of variations In mathematics, the direct method in the calculus of variations is a general method for constructing a proof of the existence of a minimizer for a given functional, introduced by Stanisław Zaremba and David Hilbert around 1900. The method reli ...
. For a non-negative integer and , the Sobolev space contains functions whose
weak derivative In mathematics, a weak derivative is a generalization of the concept of the derivative of a function (''strong derivative'') for functions not assumed differentiable, but only integrable, i.e., to lie in the L''p'' space L^1( ,b. The method ...
s of order up to are also . The inner product in is \langle f, g\rangle = \int_\Omega f(x)\bar(x)\,\mathrmx + \int_\Omega D f(x)\cdot D\bar(x)\,\mathrmx + \cdots + \int_\Omega D^s f(x)\cdot D^s \bar(x)\, \mathrmx where the dot indicates the dot product in the Euclidean space of partial derivatives of each order. Sobolev spaces can also be defined when 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 is a suitable domain, then one can define the Sobolev space as the space of
Bessel potential In mathematics, the Bessel potential is a potential (named after Friedrich Wilhelm Bessel) similar to the Riesz potential but with better decay properties at infinity. If ''s'' is a complex number with positive real part then the Bessel potentia ...
s; roughly, H^s(\Omega) = \left. \left\ \,. Here is the Laplacian and is understood in terms of the
spectral mapping theorem In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach ...
. Apart from providing a workable definition of Sobolev spaces for non-integer , this definition also has particularly desirable properties under the
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
that make it ideal for the study of pseudodifferential operators. Using these methods on a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Britis ...
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ...
, one can obtain for instance the
Hodge decomposition In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every coho ...
, which is the basis of
Hodge theory In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every co ...
.


Spaces of holomorphic functions


Hardy spaces

The
Hardy space In complex analysis, the Hardy spaces (or Hardy classes) ''Hp'' are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper . I ...
s are function spaces, arising in
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
and
harmonic analysis Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an ex ...
, whose elements are certain
holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex deriv ...
s in a complex domain. Let denote the unit disc in the complex plane. Then the Hardy space is defined as the space of holomorphic functions on such that the means M_r(f) = \frac \int_0^ \left, f\left(re^\right)\^2 \, \mathrm\theta remain bounded for . The norm on this Hardy space is defined by \left\, f\right\, _2 = \lim_ \sqrt \,. Hardy spaces in the disc are related to Fourier series. A function is in if and only if f(z) = \sum_^\infty a_n z^n where \sum_^\infty , a_n, ^2 < \infty \,. Thus consists of those functions that are ''L''2 on the circle, and whose negative frequency Fourier coefficients vanish.


Bergman spaces

The
Bergman space In complex analysis, functional analysis and operator theory, a Bergman space, named after Stefan Bergman, is a function space of holomorphic functions in a domain ''D'' of the complex plane that are sufficiently well-behaved at the boundary that t ...
s are another family of Hilbert spaces of holomorphic functions. Let be a bounded open set in the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
(or a higher-dimensional complex space) and let be the space of holomorphic functions in that are also in in the sense that \, f\, ^2 = \int_D , f(z), ^2\,\mathrm\mu(z) < \infty \,, where the integral is taken with respect to the Lebesgue measure in . Clearly is a subspace of ; in fact, it is a
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
subspace, and so a Hilbert space in its own right. This is a consequence of the estimate, valid on
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Britis ...
subsets of , that \sup_ \left, f(z)\ \le C_K \left\, f\right\, _2 \,, which in turn follows from Cauchy's integral formula. Thus convergence of a sequence of holomorphic functions in implies also
compact convergence In mathematics compact convergence (or uniform convergence on compact sets) is a type of convergence that generalizes the idea of uniform convergence. It is associated with the compact-open topology. Definition Let (X, \mathcal) be a topological ...
, and so the limit function is also holomorphic. Another consequence of this inequality is that the linear functional that evaluates a function at a point of is actually continuous on . The Riesz representation theorem implies that the evaluation functional can be represented as an element of . Thus, for every , there is a function such that f(z) = \int_D f(\zeta)\overline\,\mathrm\mu(\zeta) for all . The integrand K(\zeta, z) = \overline is known as the Bergman kernel of . This
integral kernel In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in ...
satisfies a reproducing property f(z) = \int_D f(\zeta)K(\zeta, z)\,\mathrm\mu(\zeta) \,. A Bergman space is an example of a
reproducing kernel Hilbert space In functional analysis (a branch of mathematics), a reproducing kernel Hilbert space (RKHS) is a Hilbert space of functions in which point evaluation is a continuous linear functional. Roughly speaking, this means that if two functions f and g in ...
, which is a Hilbert space of functions along with a kernel that verifies a reproducing property analogous to this one. The Hardy space also admits a reproducing kernel, known as the
Szegő kernel In the mathematical study of several complex variables, the Szegő kernel is an integral kernel that gives rise to a reproducing kernel on a natural Hilbert space of holomorphic functions. It is named for its discoverer, the Hungarian mathemat ...
. Reproducing kernels are common in other areas of mathematics as well. For instance, in
harmonic analysis Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an ex ...
the Poisson kernel is a reproducing kernel for the Hilbert space of square-integrable
harmonic function In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is, : \f ...
s in the
unit ball Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (a ...
. 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 In mathematics, an ordered basis of a vector space of finite dimension allows representing uniquely any element of the vector space by a coordinate vector, which is a sequence of scalars called coordinates. If two different bases are consider ...
from their usual finite dimensional setting. In particular, the
spectral theory In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result ...
of
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
self-adjoint In mathematics, and more specifically in abstract algebra, an element ''x'' of a *-algebra is self-adjoint if x^*=x. A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold. A collection ''C'' of elements of a st ...
linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
s on a Hilbert space generalizes the usual spectral decomposition of a
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
, and this often plays a major role in applications of the theory to other areas of mathematics and physics.


Sturm–Liouville theory

In the theory of
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
s, 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 In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
. The problem is a differential equation of the form -\frac\left (x)\frac\right+ q(x)y = \lambda w(x)y for an unknown function on an interval , satisfying general homogeneous Robin boundary conditions \begin \alpha y(a)+\alpha' y'(a) &= 0 \\ \beta y(b) + \beta' y'(b) &= 0 \,. \end The functions , , and are given in advance, and the problem is to find the function and constants for which the equation has a solution. The problem only has solutions for certain values of , called eigenvalues of the system, and this is a consequence of the spectral theorem for compact operators applied to the
integral operator An integral operator is an operator that involves integration. Special instances are: * The operator of integration itself, denoted by the integral symbol * Integral linear operators, which are linear operators induced by bilinear forms invol ...
defined by the
Green's function In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differenti ...
for the system. Furthermore, another consequence of this general result is that the eigenvalues of the system can be arranged in an increasing sequence tending to infinity.The eigenvalues of the Fredholm kernel are , which tend to zero.


Partial differential equations

Hilbert spaces form a basic tool in the study of
partial differential equations In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...
.. For many classes of partial differential equations, such as linear elliptic equations, it is possible to consider a generalized solution (known as a
weak Weak may refer to: Songs * "Weak" (AJR song), 2016 * "Weak" (Melanie C song), 2011 * "Weak" (SWV song), 1993 * "Weak" (Skunk Anansie song), 1995 * "Weak", a song by Seether from '' Seether: 2002-2013'' Television episodes * "Weak" (''Fear t ...
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 Weak formulations are important tools for the analysis of mathematical equations that permit the transfer of concepts of linear algebra to solve problems in other fields such as partial differential equations. In a weak formulation, equations or c ...
. This strategy forms the rudiment of the
Galerkin method In mathematics, in the area of numerical analysis, Galerkin methods, named after the Russian mathematician Boris Galerkin, convert a continuous operator problem, such as a differential equation, commonly in a weak formulation, to a discrete prob ...
(a
finite element method The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
) for numerical solution of partial differential equations. A typical example is the
Poisson equation Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with ...
with
Dirichlet boundary conditions In the mathematical study of differential equations, the Dirichlet (or first-type) boundary condition is a type of boundary condition, named after Peter Gustav Lejeune Dirichlet (1805–1859). When imposed on an ordinary or a partial differenti ...
in a bounded domain in . The weak formulation consists of finding a function such that, for all continuously differentiable functions in vanishing on the boundary: \int_\Omega \nabla u\cdot\nabla v = \int_\Omega gv\,. This can be recast in terms of the Hilbert space consisting of functions such that , along with its weak partial derivatives, are square integrable on , and vanish on the boundary. The question then reduces to finding in this space such that for all in this space a(u, v) = b(v) where is a continuous
bilinear form In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called '' scalars''). In other words, a bilinear form is a function that is linea ...
, and is a continuous
linear functional In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , the ...
, given respectively by a(u, v) = \int_\Omega \nabla u\cdot\nabla v,\quad b(v)= \int_\Omega gv\,. Since the Poisson equation is elliptic, it follows from Poincaré's inequality that the bilinear form 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 equation A parabolic partial differential equation is a type of partial differential equation (PDE). Parabolic PDEs are used to describe a wide variety of time-dependent phenomena, including heat conduction, particle diffusion, and pricing of derivat ...
s and certain hyperbolic partial differential equations.


Ergodic theory

The field of
ergodic theory Ergodic theory ( Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expr ...
is the study of the long-term behavior of
chaotic Chaotic was originally a Danish trading card game. It expanded to an online game in America which then became a television program based on the game. The program was able to be seen on 4Kids TV (Fox affiliates, nationwide), Jetix, The CW4Kid ...
dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...
s. The protypical case of a field that ergodic theory applies to is
thermodynamics Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws ...
, 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 The laws of thermodynamics are a set of scientific laws which define a group of physical quantities, such as temperature, energy, and entropy, that characterize thermodynamic systems in thermodynamic equilibrium. The laws also use various paramet ...
are assertions about such average behavior. In particular, one formulation of the
zeroth law of thermodynamics The zeroth law of thermodynamics is one of the four principal laws of thermodynamics. It provides an independent definition of temperature without reference to entropy, which is defined in the second law. The law was established by Ralph H. Fow ...
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 Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...
. 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 In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usuall ...
. More explicitly, suppose that the energy is fixed, and let be the subset of the phase space consisting of all states of energy (an energy surface), and let denote the evolution operator on the phase space. The dynamical system is ergodic if there are no continuous non-constant functions on such that f(T_tw) = f(w) for all on and all time . Liouville's theorem implies that there exists a measure on the energy surface that is invariant under the
time translation Time translation symmetry or temporal translation symmetry (TTS) is a mathematical transformation in physics that moves the times of events through a common interval. Time translation symmetry is the law that the laws of physics are unchanged ...
. As a result, time translation is a
unitary transformation In mathematics, a unitary transformation is a transformation that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation. Formal definition More precisely, ...
of the Hilbert space consisting of square-integrable functions on the energy surface with respect to the inner product \left\langle f, g\right\rangle_ = \int_E f\bar\,\mathrm\mu\,. The von Neumann mean ergodic theorem states the following: * If is a (strongly continuous) one-parameter semigroup of unitary operators on a Hilbert space , and is the orthogonal projection onto the space of common fixed points of , , then Px = \lim_ \frac \int_0^T U_tx\,\mathrmt\,. 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 , \underset \frac\int_0^T f(T_tw)\,\mathrmt = \int_ f(y)\,\mathrm\mu(y)\,. That is, the long time average of an observable is equal to its expectation value over an energy surface.


Fourier analysis

One of the basic goals of
Fourier analysis In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph ...
is to decompose a function into a (possibly infinite) linear combination of given basis functions: the associated
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
. The classical Fourier series associated to a function defined on the interval is a series of the form \sum_^\infty a_n e^ where a_n = \int_0^1f(\theta)e^\,\mathrm\theta\,. 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 (for integer ) shorter than the wavelength of the sawtooth itself (except for , the ''fundamental'' wave). All basis functions have nodes at the nodes of the sawtooth, but all but the fundamental have additional nodes. The oscillation of the summed terms about the sawtooth is called the Gibbs phenomenon. A significant problem in classical Fourier series asks in what sense the Fourier series converges, if at all, to the function . Hilbert space methods provide one possible answer to this question. The functions form an orthogonal basis of the Hilbert space . Consequently, any square-integrable function can be expressed as a series f(\theta) = \sum_n a_n e_n(\theta)\,,\quad a_n = \langle f, e_n\rangle and, moreover, this series converges in the Hilbert space sense (that is, in the mean). The problem can also be studied from the abstract point of view: every Hilbert space has an
orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For examp ...
, 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 . In many circumstances, it is desirable not to decompose a function into trigonometric functions, but rather into
orthogonal polynomials In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product. The most widely used orthogonal polynomials are the class ...
or
wavelet A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the num ...
s for instance, and in higher dimensions into
spherical harmonics In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form ...
. For instance, if are any orthonormal basis functions of , then a given function in can be approximated as a finite linear combination f(x) \approx f_n (x) = a_1 e_1 (x) + a_2 e_2(x) + \cdots + a_n e_n (x)\,. The coefficients are selected to make the magnitude of the difference as small as possible. Geometrically, the best approximation is the
orthogonal projection In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it wer ...
of onto the subspace consisting of all linear combinations of the , and can be calculated by a_j = \int_0^1 \overlinef (x) \, \mathrmx\,. That this formula minimizes the difference 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
eigenfunction In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
s of a
differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...
(typically the
Laplace operator In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is th ...
): this forms the foundation for the spectral study of functions, in reference to the
spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors ...
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 eigenvalue In mathematics, the Dirichlet eigenvalues are the fundamental modes of vibration of an idealized drum with a given shape. The problem of whether one can hear the shape of a drum is: given the Dirichlet eigenvalues, what features of the shape of ...
s 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 In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result ...
also underlies certain aspects of the
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
of a function. Whereas Fourier analysis decomposes a function defined on a
compact set In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i. ...
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 In physics, a continuous spectrum usually means a set of attainable values for some physical quantity (such as energy or wavelength) that is best described as an interval of real numbers, as opposed to a discrete spectrum, a set of attainable ...
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 In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' ...
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 Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an ex ...
(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

In the mathematically rigorous formulation of
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, developed by
John von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest c ...
, the possible states (more precisely, the
pure state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in t ...
s) of a quantum mechanical system are represented by
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction v ...
s (called ''state vectors'') residing in a complex separable Hilbert space, known as the
state space A state space is the set of all possible configurations of a system. It is a useful abstraction for reasoning about the behavior of a given system and is widely used in the fields of artificial intelligence and game theory. For instance, the to ...
, well defined up to a complex number of norm 1 (the phase factor). In other words, the possible states are points in the
projectivization In mathematics, projectivization is a procedure which associates with a non-zero vector space ''V'' a projective space (V), whose elements are one-dimensional subspaces of ''V''. More generally, any subset ''S'' of ''V'' closed under scalar multi ...
of a Hilbert space, usually called the
complex projective space In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a ...
. 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 In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value ...
functions, while the states for the spin of a single proton are unit elements of the two-dimensional complex Hilbert space of
spinors In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sli ...
. Each observable is represented by a
self-adjoint In mathematics, and more specifically in abstract algebra, an element ''x'' of a *-algebra is self-adjoint if x^*=x. A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold. A collection ''C'' of elements of a st ...
linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
acting on the state space. Each eigenstate of an observable corresponds to an
eigenvector In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
of the operator, and the associated
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
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 In quantum mechanics, a probability amplitude is a complex number used for describing the behaviour of systems. The modulus squared of this quantity represents a probability density. Probability amplitudes provide a relationship between the qu ...
. 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 In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
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.


Color perception

Any true physical color can be represented by a combination of pure
spectral color A spectral color is a color that is evoked by '' monochromatic light'', i.e. either a single wavelength of light in the visible spectrum, or by a relatively narrow band of wavelengths (e.g. lasers). Every wavelength of visible light is perce ...
s. 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 and in a Hilbert space are orthogonal when . The notation for this is . More generally, when is a subset in , the notation means that is orthogonal to every element from . When and are orthogonal, one has \, u + v\, ^2 = \langle u + v, u + v \rangle = \langle u, u \rangle + 2 \, \operatorname \langle u, v \rangle + \langle v, v \rangle= \, u\, ^2 + \, v\, ^2\,. By induction on , this is extended to any family of orthogonal vectors, \left\, u_1 + \cdots + u_n\right\, ^2 = \left\, u_1\right\, ^2 + \cdots + \left\, u_n\right\, ^2 . 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 of ''orthogonal'' vectors converges in if and only if the series of squares of norms converges, and \left\, \sum_^\infty u_k \right\, ^2 = \sum_^\infty \left\, u_k\right\, ^2\,. Furthermore, the sum of a series of orthogonal vectors is independent of the order in which it is taken.


Parallelogram identity and polarization

By definition, every Hilbert space is also a
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
. Furthermore, in every Hilbert space the following parallelogram identity holds: \left\, u + v\right\, ^2 + \left\, u - v\right\, ^2 = 2\left(\left\, u\right\, ^2 + \left\, v\right\, ^2\right)\,. 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 In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space. If a norm arises from an inner product the ...
. For real Hilbert spaces, the polarization identity is \langle u, v\rangle = \frac\left(\left\, u + v\right\, ^2 - \left\, u - v\right\, ^2\right)\,. For complex Hilbert spaces, it is \langle u, v\rangle = \tfrac\left(\left\, u + v\right\, ^2 - \left\, u - v\right\, ^2 + i\left\, u + iv\right\, ^2 - i\left\, u - iv\right\, ^2\right)\,. The parallelogram law implies that any Hilbert space is a
uniformly convex Banach space In mathematics, uniformly convex spaces (or uniformly rotund spaces) are common examples of reflexive Banach spaces. The concept of uniform convexity was first introduced by James A. Clarkson in 1936. Definition A uniformly convex space is a ...
.


Best approximation

This subsection employs the Hilbert projection theorem. If is a non-empty closed convex subset of a Hilbert space and a point in , there exists a unique point that minimizes the distance between and points in , y \in C \,, \quad \, x - y\, = \operatorname(x, C) = \min \\,. This is equivalent to saying that there is a point with minimal norm in the translated convex set . The proof consists in showing that every minimizing sequence is Cauchy (using the parallelogram identity) hence converges (using completeness) to a point in that has minimal norm. More generally, this holds in any uniformly convex Banach space. When this result is applied to a closed subspace of , it can be shown that the point closest to is characterized by y \in F \,, \quad x - y \perp F \,. This point is the ''orthogonal projection'' of onto , and the mapping is linear (see Orthogonal complements and projections). This result is especially significant in
applied mathematics Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathemati ...
, especially
numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods ...
, where it forms the basis of
least squares The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the re ...
methods. In particular, when is not equal to , one can find a nonzero vector orthogonal to (select and ). A very useful criterion is obtained by applying this observation to the closed subspace generated by a subset of . : A subset of spans a dense vector subspace if (and only if) the vector 0 is the sole vector orthogonal to .


Duality

The
dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by cons ...
is the space of all
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
linear functions from the space into the base field. It carries a natural norm, defined by \, \varphi\, = \sup_ , \varphi(x), \,. 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 In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space. If a norm arises from an inner product the ...
. The dual space is also complete so it is a Hilbert space in its own right. If is a complete orthonormal basis for then the inner product on the dual space of any two f, g \in H^* is \langle f, g \rangle_ = \sum_ f (e_i) \overline where all but countably many of the terms in this series are zero. The
Riesz representation theorem :''This article describes a theorem concerning the dual of a Hilbert space. For the theorems relating linear functionals to Measure (mathematics), measures, see Riesz–Markov–Kakutani representation theorem.'' The Riesz representation theorem, ...
affords a convenient description of the dual space. To every element of , there is a unique element of , defined by \varphi_u(x) = \langle x, u\rangle where moreover, \left\, \varphi_u \right\, = \left\, u \right\, . The Riesz representation theorem states that the map from to defined by is
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element o ...
, which makes this map an isometric
antilinear In mathematics, a function f : V \to W between two complex vector spaces is said to be antilinear or conjugate-linear if \begin f(x + y) &= f(x) + f(y) && \qquad \text \\ f(s x) &= \overline f(x) && \qquad \text \\ \end hold for all vectors x, y ...
isomorphism. So to every element of the dual there exists one and only one in such that \langle x, u_\varphi\rangle = \varphi(x) for all . The inner product on the dual space satisfies \langle \varphi, \psi \rangle = \langle u_\psi, u_\varphi \rangle \,. The reversal of order on the right-hand side restores linearity in from the antilinearity of . In the real case, the antilinear isomorphism from to its dual is actually an isomorphism, and so real Hilbert spaces are naturally isomorphic to their own duals. The representing vector is obtained in the following way. When , the
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learn ...
is a closed vector subspace of , not equal to , hence there exists a nonzero vector orthogonal to . The vector is a suitable scalar multiple of . The requirement that yields u = \langle v, v \rangle^ \, \overline \, v \,. This correspondence is exploited by the
bra–ket notation In quantum mechanics, bra–ket notation, or Dirac notation, is used ubiquitously to denote quantum states. The notation uses angle brackets, and , and a vertical bar , to construct "bras" and "kets". A ket is of the form , v \rangle. Mathem ...
popular in
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
. It is common in physics to assume that the inner product, denoted by , is linear on the right, \langle x , y \rangle = \langle y, x \rangle \,. The result can be seen as the action of the linear functional (the ''bra'') on the vector (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 is reflexive, meaning that the natural map from into its double dual space is an isomorphism.


Weakly-convergent sequences

In a Hilbert space , a sequence is weakly convergent to a vector when \lim_n \langle x_n, v \rangle = \langle x, v \rangle for every . For example, any orthonormal sequence converges weakly to 0, as a consequence of Bessel's inequality. Every weakly convergent sequence is bounded, by the
uniform boundedness principle In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the corner ...
. 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 function In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigraph (the set of poi ...
als, in the same way that the Bolzano–Weierstrass theorem is used for continuous functions on . Among several variants, one simple statement is as follows: :If is a convex continuous function such that tends to when tends to , then admits a minimum at some point . This fact (and its various generalizations) are fundamental for direct methods in the
calculus of variations The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
. 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 are weakly compact, since is reflexive. The existence of weakly convergent subsequences is a special case of the
Eberlein–Šmulian theorem In the mathematical field of functional analysis, the Eberlein–Šmulian theorem (named after William Frederick Eberlein and Witold Lwowitsch Schmulian) is a result that relates three different kinds of weak compactness in a Banach space. Stat ...
.


Banach space properties

Any general property of
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
s continues to hold for Hilbert spaces. The open mapping theorem states that a
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element o ...
linear transformation from one Banach space to another is an
open mapping In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y. Likewise, ...
meaning that it sends open sets to open sets. A corollary is the bounded inverse theorem, that a continuous and
bijective In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
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 In mathematics, the closed graph theorem may refer to one of several basic results characterizing continuous functions in terms of their graphs. Each gives conditions when functions with closed graphs are necessarily continuous. Graphs and m ...
, 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 geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a ...
. In the case of Hilbert spaces, this is basic in the study of
unbounded operator In mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases. The ter ...
s (see closed operator). The (geometrical)
Hahn–Banach theorem The Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear f ...
asserts that a closed convex set can be separated from any point outside it by means of a
hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its '' ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...
of the Hilbert space. This is an immediate consequence of the best approximation property: if is the element of a closed convex set closest to , then the separating hyperplane is the plane perpendicular to the segment passing through its midpoint.


Operators on Hilbert spaces


Bounded operators

The
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
s from a Hilbert space to a second Hilbert space are ''bounded'' in the sense that they map
bounded set :''"Bounded" and "boundary" are distinct concepts; for the latter see boundary (topology). A circle in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary. In mathematical analysis and related areas of m ...
s to bounded sets. Conversely, if an operator is bounded, then it is continuous. The space of such
bounded linear operator In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vect ...
s has a norm, the
operator norm In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces. Introd ...
given by \lVert A \rVert = \sup \left\\,. The sum and the composite of two bounded linear operators is again bounded and linear. For ''y'' in ''H''2, the map that sends to is linear and continuous, and according to the
Riesz representation theorem :''This article describes a theorem concerning the dual of a Hilbert space. For the theorems relating linear functionals to Measure (mathematics), measures, see Riesz–Markov–Kakutani representation theorem.'' The Riesz representation theorem, ...
can therefore be represented in the form \left\langle x, A^* y \right\rangle = \langle Ax, y \rangle for some vector in . This defines another bounded linear operator , the adjoint of . The adjoint satisfies . When the Riesz representation theorem is used to identify each Hilbert space with its continuous dual space, the adjoint of can be shown to be identical to the
transpose In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The tr ...
of , which by definition sends \psi \in H_2^ to the functional \psi \circ A \in H_1^. The set of all bounded linear operators on (meaning operators ), together with the addition and composition operations, the norm and the adjoint operation, is a
C*-algebra In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuou ...
, which is a type of
operator algebra In functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with the multiplication given by the composition of mappings. The results obtained in the study of ...
. An element of is called 'self-adjoint' or 'Hermitian' if . If is Hermitian and for every , then is called 'nonnegative', written ; if equality holds only when , then is called 'positive'. The set of self adjoint operators admits a
partial order In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
, in which if . If has the form for some , then is nonnegative; if is invertible, then is positive. A converse is also true in the sense that, for a non-negative operator , there exists a unique non-negative
square root In mathematics, a square root of a number is a number such that ; in other words, a number whose '' square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . ...
such that A = B^2 = B^*B\,. In a sense made precise by the
spectral theorem In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful be ...
, self-adjoint operators can usefully be thought of as operators that are "real". An element of is called ''normal'' if . Normal operators decompose into the sum of a self-adjoint operator and an imaginary multiple of a self adjoint operator A = \frac + i\frac that commute with each other. Normal operators can also usefully be thought of in terms of their real and imaginary parts. An element of is called
unitary Unitary may refer to: Mathematics * Unitary divisor * Unitary element * Unitary group * Unitary matrix * Unitary morphism * Unitary operator * Unitary transformation * Unitary representation In mathematics, a unitary representation of a grou ...
if is invertible and its inverse is given by . This can also be expressed by requiring that be onto and for all . The unitary operators form a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
under composition, which is the
isometry group In mathematics, the isometry group of a metric space is the set of all bijective isometries (i.e. bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation. Its identity element is the ...
of . An element of is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Britis ...
if it sends bounded sets to
relatively compact In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact. Properties Every subset of a compact topological space is relatively compact (sinc ...
sets. Equivalently, a bounded operator is compact if, for any bounded sequence , the sequence has a convergent subsequence. Many
integral operator An integral operator is an operator that involves integration. Special instances are: * The operator of integration itself, denoted by the integral symbol * Integral linear operators, which are linear operators induced by bilinear forms invol ...
s are compact, and in fact define a special class of operators known as
Hilbert–Schmidt operator In mathematics, a Hilbert–Schmidt operator, named after David Hilbert and Erhard Schmidt, is a bounded operator A \colon H \to H that acts on a Hilbert space H and has finite Hilbert–Schmidt norm \, A\, ^2_ \ \stackrel\ \sum_ \, Ae_i\, ...
s 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 Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learn ...
and
cokernel The cokernel of a linear mapping of vector spaces is the quotient space of the codomain of by the image of . The dimension of the cokernel is called the ''corank'' of . Cokernels are dual to the kernels of category theory, hence the nam ...
. The index of a Fredholm operator is defined by \operatorname T = \dim\ker T - \dim\operatorname T \,. The index is
homotopy In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deform ...
invariant, and plays a deep role in
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
via the
Atiyah–Singer index theorem In differential geometry, the Atiyah–Singer index theorem, proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold, the analytical index (related to the dimension of the sp ...
.


Unbounded operators

Unbounded operator In mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases. The ter ...
s are also tractable in Hilbert spaces, and have important applications to
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
. An unbounded operator on a Hilbert space is defined as a linear operator whose domain is a linear subspace of . Often the domain is a dense subspace of , in which case 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 are: * A suitable extension of the differential operator (A f)(x) = -i \frac f(x) \,, where is the imaginary unit and is a differentiable function of compact support. * The multiplication-by- operator: (B f) (x) = x f(x)\,. These correspond to the
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...
and position observables, respectively. Note that neither nor is defined on all of , since in the case of the derivative need not exist, and in the case of the product function need not be square integrable. In both cases, the set of possible arguments form dense subspaces of .


Constructions


Direct sums

Two Hilbert spaces and can be combined into another Hilbert space, called the (orthogonal) direct sum, and denoted H_1 \oplus H_2 \,, consisting of the set of all
ordered pair In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...
s where , , and inner product defined by \bigl\langle (x_1, x_2), (y_1, y_2)\bigr\rangle_ = \left\langle x_1, y_1\right\rangle_ + \left\langle x_2, y_2\right\rangle_ \,. More generally, if is a family of Hilbert spaces indexed by , then the direct sum of the , denoted \bigoplus_H_i consists of the set of all indexed families x = (x_i \in H_i, i \in I) \in \prod_H_i in the
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\t ...
of the such that \sum_ \, x_i\, ^2 < \infty \,. The inner product is defined by \langle x, y\rangle = \sum_ \left\langle x_i, y_i\right\rangle_ \,. Each of the is included as a closed subspace in the direct sum of all of the . Moreover, the are pairwise orthogonal. Conversely, if there is a system of closed subspaces, , , in a Hilbert space , that are pairwise orthogonal and whose union is dense in , then is canonically isomorphic to the direct sum of . In this case, is called the internal direct sum of the . A direct sum (internal or external) is also equipped with a family of orthogonal projections onto the th direct summand . These projections are bounded, self-adjoint,
idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
operators that satisfy the orthogonality condition E_i E_j = 0,\quad i \neq j \,. The
spectral theorem In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful be ...
for
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Britis ...
self-adjoint operators on a Hilbert space states that 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 The Fock space is an algebraic construction used in quantum mechanics to construct the quantum states space of a variable or unknown number of identical particles from a single particle Hilbert space . It is named after V. A. Fock who first intr ...
of a system containing a variable number of particles, where each Hilbert space in the direct sum corresponds to an additional
degree of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
for the quantum mechanical system. In
representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
, the
Peter–Weyl theorem In mathematics, the Peter–Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian. It was initially proved by Hermann Weyl, with his student Fritz Peter, ...
guarantees that any
unitary representation In mathematics, a unitary representation of a group ''G'' is a linear representation π of ''G'' on a complex Hilbert space ''V'' such that π(''g'') is a unitary operator for every ''g'' ∈ ''G''. The general theory is well-developed in case ''G ...
of a
compact group In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural gen ...
on a Hilbert space splits as the direct sum of finite-dimensional representations.


Tensor products

If and , then one defines an inner product on the (ordinary)
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
as follows. On simple tensors, let \langle x_1 \otimes x_2, \, y_1 \otimes y_2 \rangle = \langle x_1, y_1 \rangle \, \langle x_2, y_2 \rangle \,. This formula then extends by sesquilinearity to an inner product on . The Hilbertian tensor product of and , sometimes denoted by , is the Hilbert space obtained by completing for the metric associated to this inner product. An example is provided by the Hilbert space . The Hilbertian tensor product of two copies of is isometrically and linearly isomorphic to the space of square-integrable functions on the square . This isomorphism sends a simple tensor to the function (s, t) \mapsto f_1(s) \, f_2(t) on the square. This example is typical in the following sense. Associated to every simple tensor product is the rank one operator from to that maps a given as x^* \mapsto x^*(x_1) x_2 \,. This mapping defined on simple tensors extends to a linear identification between and the space of finite rank operators from to . This extends to a linear isometry of the Hilbertian tensor product with the Hilbert space of
Hilbert–Schmidt operator In mathematics, a Hilbert–Schmidt operator, named after David Hilbert and Erhard Schmidt, is a bounded operator A \colon H \to H that acts on a Hilbert space H and has finite Hilbert–Schmidt norm \, A\, ^2_ \ \stackrel\ \sum_ \, Ae_i\, ...
s from to .


Orthonormal bases

The notion of an
orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For examp ...
from linear algebra generalizes over to the case of Hilbert spaces. In a Hilbert space , an orthonormal basis is a family of elements of satisfying the conditions: # ''Orthogonality'': Every two different elements of are orthogonal: for all with . # ''Normalization'': Every element of the family has norm 1: for all . # ''Completeness'': The
linear span In mathematics, the linear span (also called the linear hull or just span) of a set of vectors (from a vector space), denoted , pp. 29-30, §§ 2.5, 2.8 is defined as the set of all linear combinations of the vectors in . It can be characterized ...
of the family , , 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 is
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
). Such a system is always
linearly independent In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
. 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 , if for all , then . This is related to the fact that the only vector orthogonal to a dense linear subspace is the zero vector, for if is any orthonormal set and is orthogonal to , then is orthogonal to the closure of the linear span of , which is the whole space. Examples of orthonormal bases include: * the set forms an orthonormal basis of with the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alg ...
; * the sequence with forms an orthonormal basis of the complex space ; In the infinite-dimensional case, an orthonormal basis will not be a basis in the sense of
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrice ...
; 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 \ell_2 of square-summable sequences of complex numbers is the set of infinite sequences (c_1, c_2, c_3, \dots) of real or complex numbers such that \left, c_1\^2 + \left, c_2\^2 + \left, c_3\^2 + \cdots < \infty \,. This space has an orthonormal basis: \begin e_1 &= (1, 0, 0, \dots) \\ e_2 &= (0, 1, 0, \dots) \\ & \ \ \vdots \end This space is the infinite-dimensional generalization of the \ell_2^n space of finite-dimensional vectors. It is usually the first example used to show that in infinite-dimensional spaces, a set that is
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
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 \ell_2 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 \ell_2 in many ways. For example, if is any (infinite) set, then one can form a Hilbert space of sequences with index set , defined by \ell^2(B) =\left\ \,. The summation over ''B'' is here defined by \sum_ \left, x (b)\^2 = \sup \sum_^N \left, x(b_n)\^2 the
supremum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest ...
being taken over all finite subsets of . It follows that, for this sum to be finite, every element of has only countably many nonzero terms. This space becomes a Hilbert space with the inner product \langle x, y \rangle = \sum_ x(b)\overline for all . Here the sum also has only countably many nonzero terms, and is unconditionally convergent by the Cauchy–Schwarz inequality. An orthonormal basis of is indexed by the set , given by :e_b(b') = \begin 1 & \text b=b'\\ 0 & \text \end


Bessel's inequality and Parseval's formula

Let be a finite orthonormal system in . For an arbitrary vector , let y = \sum_^n \langle x, f_j \rangle \, f_j \,. Then for every . It follows that is orthogonal to each , hence is orthogonal to . Using the Pythagorean identity twice, it follows that \, x\, ^2 = \, x - y\, ^2 + \, y\, ^2 \ge \, y\, ^2 = \sum_^n\bigl, \langle x, f_j \rangle\bigr, ^2 \,. Let , be an arbitrary orthonormal system in . Applying the preceding inequality to every finite subset of gives Bessel's inequality: \sum_\bigl, \langle x, f_i \rangle\bigr, ^2 \le \, x\, ^2, \quad x \in H (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 onto the linear subspace spanned by the has norm that does not exceed that of . 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 is an orthonormal basis of , then every element of may be written as x = \sum_ \left\langle x, e_k \right\rangle \, e_k \,. Even if 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 , and the individual coefficients are the Fourier coefficients of . Parseval's identity then asserts that \, x\, ^2 = \sum_, \langle x, e_k\rangle, ^2 \,. Conversely, if is an orthonormal set such that Parseval's identity holds for every , then 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 In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
, called the Hilbert dimension of the space. For instance, since has an orthonormal basis indexed by , its Hilbert dimension is the cardinality of (which may be a finite integer, or a countable or uncountable
cardinal number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. ...
). The Hilbert dimension is not greater than the
Hamel dimension In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to di ...
(the usual dimension of a vector space). The two dimensions are equal if and only one is finite. As a consequence of Parseval's identity, if is an orthonormal basis of , then the map defined by is an isometric isomorphism of Hilbert spaces: it is a bijective linear mapping such that \bigl\langle \Phi (x), \Phi(y) \bigr\rangle_ = \left\langle x, y \right\rangle_H for all . The
cardinal number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. ...
of is the Hilbert dimension of . Thus every Hilbert space is isometrically isomorphic to a sequence space for some set .


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 In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
orthonormal basis. All infinite-dimensional separable Hilbert spaces are therefore isometrically isomorphic to . In the past, Hilbert spaces were often required to be separable as part of the definition. 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 In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, most of the Hilbert spaces are in fact separable, as stipulated by the
Wightman axioms In mathematical physics, the Wightman axioms (also called Gårding–Wightman axioms), named after Arthur Wightman, are an attempt at a mathematically rigorous formulation of quantum field theory. Arthur Wightman formulated the axioms in the ear ...
. 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 Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
and any infinite Hilbert tensor product (of spaces of dimension greater than one) is non-separable. For instance, a
bosonic field In quantum field theory, a bosonic field is a quantum field whose quanta are bosons; that is, they obey Bose–Einstein statistics. Bosonic fields obey canonical commutation relations, as distinct from the canonical anticommutation relations obeye ...
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 is a subset of a Hilbert space , the set of vectors orthogonal to is defined by S^\perp = \left\ \,. The set is a
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
subspace of (can be proved easily using the linearity and continuity of the inner product) and so forms itself a Hilbert space. If is a closed subspace of , then is called the of . In fact, every can then be written uniquely as , with and . Therefore, is the internal Hilbert direct sum of and . The linear operator that maps to is called the onto . There is a
natural Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans ar ...
one-to-one correspondence between the set of all closed subspaces of and the set of all bounded self-adjoint operators such that . Specifically, This provides the geometrical interpretation of : it is the best approximation to ''x'' by elements of ''V''. Projections and are called mutually orthogonal if . This is equivalent to and being orthogonal as subspaces of . The sum of the two projections and is a projection only if and are orthogonal to each other, and in that case . The composite is generally not a projection; in fact, the composite is a projection if and only if the two projections commute, and in that case . By restricting the codomain to the Hilbert space , the orthogonal projection gives rise to a projection mapping ; it is the adjoint of the
inclusion mapping In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function \iota that sends each element x of A to x, treated as an element of B: \iota : A\rightarrow B, \qquad \iot ...
i : V \to H \,, meaning that \left\langle i x, y\right\rangle_H = \left\langle x, \pi y \right\rangle_V for all and . The operator norm of the orthogonal projection onto a nonzero closed subspace is equal to 1: \, P_V\, = \sup_ \frac = 1 \,. Every closed subspace ''V'' of a Hilbert space is therefore the image of an operator of norm one such that . 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 , there is an operator of norm one whose image is such that . While this result characterizes the metric structure of a Hilbert space, the structure of a Hilbert space as a
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
can itself be characterized in terms of the presence of complementary subspaces: * A Banach space is topologically and linearly isomorphic to a Hilbert space if and only if, to every closed subspace , there is a closed subspace such that is equal to the internal direct sum . The orthogonal complement satisfies some more elementary results. It is a
monotone function In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...
in the sense that if , then with equality holding if and only if is contained in the closure of . This result is a special case of the
Hahn–Banach theorem The Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear f ...
. The closure of a subspace can be completely characterized in terms of the orthogonal complement: if is a subspace of , then the closure of is equal to . The orthogonal complement is thus a
Galois connection In mathematics, especially in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets). Galois connections find applications in various mathematical theories. They generalize the funda ...
on the
partial order In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
of subspaces of a Hilbert space. In general, the orthogonal complement of a sum of subspaces is the intersection of the orthogonal complements: \left(\sum_i V_i\right)^\perp = \bigcap_i V_i^\perp \,. If the are in addition closed, then \overline = \left(\bigcap_i V_i\right)^\perp \,.


Spectral theory

There is a well-developed
spectral theory In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result ...
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 , denoted , is the set of complex numbers such that lacks a continuous inverse. If is bounded, then the spectrum is always a
compact set In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i. ...
in the complex plane, and lies inside the disc . If is self-adjoint, then the spectrum is real. In fact, it is contained in the interval where m = \inf_\langle Tx, x\rangle \,,\quad M = \sup_\langle Tx, x\rangle \,. Moreover, and are both actually contained within the spectrum. The eigenspaces of an operator are given by H_\lambda = \ker(T - \lambda)\,. Unlike with finite matrices, not every element of the spectrum of must be an eigenvalue: the linear operator 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 In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful be ...
of a self-adjoint operator takes a particularly simple form if, in addition, is assumed to be a compact operator. The spectral theorem for compact self-adjoint operators states: * A compact self-adjoint operator has only countably (or finitely) many spectral values. The spectrum of has no
limit point In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contai ...
in the complex plane except possibly zero. The eigenspaces of decompose into an orthogonal direct sum: H = \bigoplus_H_\lambda \,. Moreover, if denotes the orthogonal projection onto the eigenspace , then T = \sum_ \lambda E_\lambda \,, where the sum converges with respect to the norm on . 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 operator In mathematics, a Hilbert–Schmidt operator, named after David Hilbert and Erhard Schmidt, is a bounded operator A \colon H \to H that acts on a Hilbert space H and has finite Hilbert–Schmidt norm \, A\, ^2_ \ \stackrel\ \sum_ \, Ae_i\, ...
s. The general spectral theorem for self-adjoint operators involves a kind of operator-valued
Riemann–Stieltjes integral In mathematics, the Riemann–Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. The definition of this integral was first published in 1894 by Stieltjes. It serves as an inst ...
, rather than an infinite summation. The ''spectral family'' associated to associates to each real number λ an operator , which is the projection onto the nullspace of the operator , where the positive part of a self-adjoint operator is defined by A^+ = \tfrac\left(\sqrt + A\right) \,. The operators 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 T = \int_\mathbb \lambda\, \mathrmE_\lambda \,. The integral is understood as a Riemann–Stieltjes integral, convergent with respect to the norm on . In particular, one has the ordinary scalar-valued integral representation \langle Tx, y\rangle = \int_ \lambda\,\mathrm\langle E_\lambda x, y\rangle \,. 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 must instead be replaced by a
resolution of the identity In functional analysis, a branch of mathematics, the Borel functional calculus is a '' functional calculus'' (that is, an assignment of operators from commutative algebras to functions defined on their spectra), which has particularly broad scop ...
. A major application of spectral methods is the
spectral mapping theorem In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach ...
, which allows one to apply to a self-adjoint operator any continuous complex function defined on the spectrum of by forming the integral f(T) = \int_ f(\lambda)\,\mathrmE_\lambda \,. The resulting
continuous functional calculus In mathematics, particularly in operator theory and C*-algebra theory, a continuous functional calculus is a functional calculus which allows the application of a continuous function to normal elements of a C*-algebra. Theorem Theorem. Let ...
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: is a spectral value if the resolvent operator R_\lambda = (T - \lambda)^ fails to be a well-defined continuous operator. The self-adjointness of still guarantees that the spectrum is real. Thus the essential idea of working with unbounded operators is to look instead at the resolvent where is nonreal. This is a ''bounded'' normal operator, which admits a spectral representation that can then be transferred to a spectral representation of 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 In mathematics, the Bessel potential is a potential (named after Friedrich Wilhelm Bessel) similar to the Riesz potential but with better decay properties at infinity. If ''s'' is a complex number with positive real part then the Bessel potentia ...
. A precise version of the spectral theorem in this case is: There is also a version of the spectral theorem that applies to unbounded normal operators.


In popular culture

In ''
Gravity's Rainbow ''Gravity's Rainbow'' is a 1973 novel by American writer Thomas Pynchon. The narrative is set primarily in Europe at the end of World War II and centers on the design, production and dispatch of V-2 rockets by the German military. In particular, ...
'' (1973), a novel by
Thomas Pynchon Thomas Ruggles Pynchon Jr. ( , ; born May 8, 1937) is an American novelist noted for his dense and complex novels. His fiction and non-fiction writings encompass a vast array of subject matter, genres and themes, including history, music, scie ...
, one of the characters is called "Sammy Hilbert-Spaess", a pun on "Hilbert Space". The novel refers also to
Gödel's incompleteness theorems Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the phil ...
.


See also

* * * * * * * * * * * * * *


Remarks


Notes


References

* . * . * . * . * . * . * . * . * . * . * . * . * . * . * . * . * . * . * . * * . * . * . * . * . * . * * . * . * . * . * . *. * . * . * . * . * . * * . *. * . * * * . * . * . * . * . * . * . * * . * ; originally published ''Monografje Matematyczne'', vol. 7, Warszawa, 1937. * * . * . * . * . * . * . * . * . * . * . * . * . * . * .


External links

*
Hilbert space at Mathworld

245B, notes 5: Hilbert spaces
by
Terence Tao Terence Chi-Shen Tao (; born 17 July 1975) is an Australian-American mathematician. He is a professor of mathematics at the University of California, Los Angeles (UCLA), where he holds the James and Carol Collins chair. His research includes ...
{{good article Functional analysis Linear algebra Operator theory
Space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consi ...