TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, Hilbert spaces (named for
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, G ...
) 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 matrix (mat ...
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 generalizations ...

from the two-dimensional and three dimensional
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
s to spaces that may have an infinite
dimension In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

. A Hilbert space is a
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
equipped with an
inner product In mathematics, an inner product space or a Hausdorff space, Hausdorff pre-Hilbert space is a vector space with a binary operation called an inner product. This operation associates each pair of vectors in the space with a Scalar (mathematics), ...
operation, which allows defining a
distance function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

and
perpendicularity In elementary geometry, the property of being perpendicular (perpendicularity) is the relationship between two Line (geometry), lines which meet at a right angle (90 Degree (angle), degrees). The property extends to other related Mathematical ob ...
(known as
orthogonality In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

in this context). Furthermore, Hilbert spaces are complete for this distance, which means that there are enough
limits Limit or Limits may refer to: Arts and media * Limit (music) In music theory, limit or harmonic limit is a way of characterizing the harmony found in a piece or genre (music), genre of music, or the harmonies that can be made using a particular ...
in the space to allow the techniques of calculus to be used. Hilbert spaces arise naturally and frequently in mathematics and
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

, typically as
infinite-dimensional In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
function space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s. 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 This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, G ...
,
Erhard Schmidt Erhard Schmidt (13 January 1876 – 6 December 1959) was a Baltic German The Baltic Germans (german: Deutsch-Balten or , later ; and остзейцы ''ostzeitsy'' 'Balters' in Russian) are ethnic German inhabitants of the eastern shores ...

, and
Frigyes Riesz Frigyes Riesz ( hu, Riesz Frigyes, , sometimes spelled as Frederic; 22 January 1880 – 28 February 1956) was a HungarianEberhard Zeidler: Nonlinear Functional Analysis and Its Applications: Linear monotone operators. Springer, 199/ref> mathematic ...
. They are indispensable tools in the theories of
partial differential equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
s,
quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...
,
Fourier analysis In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis ...
(which includes applications to
signal processing Signal processing is an electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetis ...

and heat transfer), and
ergodic theory Ergodic theory ( Greek: ' "work", ' "way") is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
(which forms the mathematical underpinning of
thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...

).
John von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian Americans, Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. Von Neumann was generally rega ...

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 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional analysis is a branch of mathemat ...
. Apart from the classical Euclidean 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 normed space, norm that is a combination of Lp norm, ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a s ...
s consisting of
generalized function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no genera ...
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 A rectangular grid (top) and its image under a conformal map ''f'' (bottom). In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algeb ...
s. Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the
Pythagorean theorem In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

and
parallelogram law A parallelogram. The sides are shown in blue and the diagonals in red. In mathematics, the simplest form of the parallelogram law (also called the parallelogram identity) belongs to elementary geometry. It states that the sum of the squares of the ...

hold in a Hilbert space. At a deeper level, perpendicular projection onto 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 a set of
coordinate axes A Cartesian coordinate system (, ) in a plane is a coordinate system In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the o ...
(an
orthonormal basisIn linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of u ...
), in analogy with
Cartesian coordinates A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft Arts, entertainment and media *Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons'') ...

in the plane. When that set of axes is
countably infinite In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, the Hilbert space can also be usefully thought of in terms of the space of
infinite sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
s that are square-summable. The latter space is often in the older literature referred to as ''the'' Hilbert space.
Linear operator In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
s on a Hilbert space are likewise fairly concrete objects: in good cases, they are simply transformations that stretch the space by different factors in mutually perpendicular directions in a sense that is made precise by the study of their
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 Continuum may refer to: * Continuum (measurement) Continuum theories or models expla ...
.

# 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 classical geometry. Originally, it was the three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ...
consisting of three-dimensional
vectors Vector may refer to: Biology *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism; a disease vector *Vector (molecular biology), a DNA molecule used as a vehicle to artificially carr ...
, denoted by , and equipped with the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...
. 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 Plane or planes may refer to: * Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft Arts, entertainment and media *Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons'') ...

, 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 Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out se ...

in its first argument: for any scalars , , and vectors , , and . # It is
positive definiteIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
: for all vectors , , with equality
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
. 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 a Hausdorff space, Hausdorff pre-Hilbert space is a vector space with a binary operation called an inner product. This operation associates each pair of vectors in the space with a Scalar (mathematics), ...
. A
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
equipped with such an inner product is known as a (real)
inner product space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
. 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 Norm, the Norm or NORM may refer to: In academic disciplines * Norm (geology), an estimate of the idealised mineral content of a rock * Norm (philosophy) Norms are concepts ( sentences) of practical import, oriented to effecting an action, rat ...
) 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 variables, functions of several variables: the Differential calculus, different ...
in Euclidean space relies on the ability to compute
limits Limit or Limits may refer to: Arts and media * Limit (music) In music theory, limit or harmonic limit is a way of characterizing the harmony found in a piece or genre (music), genre of music, or the harmonies that can be made using a particular ...
, and to have useful criteria for concluding that limits exist. A
mathematical series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
$\sum_^\infty \mathbf_n$ consisting of vectors in is
absolutely convergent In mathematics, an Series (mathematics), 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 number, real or Complex number, ...
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 contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

s. The
complex plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
denoted by is equipped with a notion of magnitude, the which is defined as the square root of the product of with its
complex conjugate In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
: $, z, ^2 = z\overline \,.$ 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'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...
. 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: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
or
complex The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public university, Public rese ...

inner product space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
that is also a
complete metric space In mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathema ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
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 Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out se ...
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 definiteIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
: $\begin \langle x, x\rangle > 0 & \quad \text x \neq 0, \\ \langle x, x\rangle = 0 & \quad \text x = 0\,. \end$ 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
and $\left(H, H, \langle \cdot, \cdot \rangle\right)$ will form a
dual system In the field of functional analysis Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction i ...
. The
norm Norm, the Norm or NORM may refer to: In academic disciplines * Norm (geology), an estimate of the idealised mineral content of a rock * Norm (philosophy) Norms are concepts ( sentences) of practical import, oriented to effecting an action, rat ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, 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 at least one of the vectors in the set can be defined as a linear combinationIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics ...
. With a distance function defined in this way, any inner product space is a
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
, 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 or, alternatively, if every Cauchy sequence in converges in . Intuitively, a space is complet ...
is a Hilbert space. The of is expressed using a form of the Cauchy criterion for sequences in : a pre-Hilbert space is complete if every
Cauchy sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
s. As such they are
topological vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s, in which
topological s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

notions like the
openness Openness is an overarching concept or philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, Epistemology, knowledge, Ethics, values, Philosophy of mind, mi ...
and closedness of subsets are well defined. Of special importance is the notion of a closed
linear subspace In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
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 Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functiona ...
consists of all infinite sequences of complex numbers such that the
series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used i ...
$\sum_^\infty , z_n, ^2$ converges. The inner product on is defined by $\langle \mathbf, \mathbf\rangle = \sum_^\infty z_n\overline\,,$ with the latter series converging as a consequence of the
Cauchy–Schwarz inequality In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
. 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 Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathematics), series, and analytic ...
, 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...

s and physicists. In particular, the idea of an abstract linear space (vector space) had gained some traction towards the end of the 19th century: this is a space whose elements can be added together and multiplied by scalars (such as
real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...

or
complex numbers In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
) 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
(including
series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used i ...
) 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 This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, G ...
and
Erhard Schmidt Erhard Schmidt (13 January 1876 – 6 December 1959) was a Baltic German The Baltic Germans (german: Deutsch-Balten or , later ; and остзейцы ''ostzeitsy'' 'Balters' in Russian) are ethnic German inhabitants of the eastern shores ...

's study of
integral equations In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, that two
square-integrable In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
real-valued functions and on an interval have an ''inner product'' : $\langle f, g \rangle = \int_a^b f\left(x\right)g\left(x\right)\, \mathrmx$ 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 notion of perpendicularity to the linear algebra of bilinear forms. Two elements ''u'' and ''v'' of a vector space with bilinear form ''B'' are orthogonal when . Depending on the bili ...

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 : $f\left(x\right) \mapsto \int_a^b K\left(x, y\right) f\left(y\right)\, \mathrmy$ where is a continuous function symmetric in and . The resulting eigenfunction expansion expresses the function as a series of the form : $K\left(x, y\right) = \sum_n \lambda_n\varphi_n\left(x\right)\varphi_n\left(y\right)$ 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
, an alternative to the
Riemann integral The partition does not need to be regular, as shown here. The approximation works as long as the width of each subdivision tends to zero. In the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such to ...
introduced by
Henri Lebesgue Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (a ...
in 1904. The Lebesgue integral made it possible to integrate a much broader class of functions. In 1907,
Frigyes Riesz Frigyes Riesz ( hu, Riesz Frigyes, , sometimes spelled as Frederic; 22 January 1880 – 28 February 1956) was a HungarianEberhard Zeidler: Nonlinear Functional Analysis and Its Applications: Linear monotone operators. Springer, 199/ref> mathematic ...
and
Ernst Sigismund Fischer Ernst Sigismund Fischer (12 July 1875 – 14 November 1954) was a mathematician born in Vienna, Austria en, Viennese , iso_code = AT-9 , registration_plate = W , postal_code_type = P ...
independently proved that the space of square Lebesgue-integrable functions is a
complete metric space In mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathema ...
. 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 French (french: français(e), link=no) may refer to: * Something of, from, or related to France France (), officially the French Republic (french: link=no, R ...

,
Friedrich Bessel Friedrich Wilhelm Bessel (; 22 July 1784 – 17 March 1846) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citizens of Germa ...
and Marc-Antoine Parseval on
trigonometric series In mathematics, a trigonometric series is a series of the form: : \frac+\displaystyle\sum_^(A_ \cos + B_ \sin). It is called a Fourier series if the terms A_ and B_ have the form: :A_=\frac\displaystyle\int^_0\! f(x) \cos \,dx\qquad (n=0,1,2, ...
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 The mathematical concept of a Hilbert space, named after David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and one o ...
was independently established by
Maurice FréchetMaurice may refer to: People *Saint Maurice Saint Maurice (also Moritz, Morris, or Mauritius; ) was the leader of the legendary Roman Theban Legion in the 3rd century, and one of the favorite and most widely venerated saints of that group. He w ...
and
Frigyes Riesz Frigyes Riesz ( hu, Riesz Frigyes, , sometimes spelled as Frederic; 22 January 1880 – 28 February 1956) was a HungarianEberhard Zeidler: Nonlinear Functional Analysis and Its Applications: Linear monotone operators. Springer, 199/ref> mathematic ...
in 1907.
John von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian Americans, Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. Von Neumann was generally rega ...

coined the term ''abstract Hilbert space'' in his work on unbounded Hermitian operators. Although other mathematicians such as
Hermann Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citizens of ...

and
Norbert Wiener Norbert Wiener (November 26, 1894 – March 18, 1964) was an American mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...

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 and also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his cont ...
. 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s on that space, the
symmetries Symmetry (from Ancient Greek, Greek συμμετρία ''symmetria'' "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" ...

of the system are
unitary operator In functional analysis Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional ...
s, and
measurements Measurement is the quantification (science), quantification of variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or events. The scope and application of measurement are dependent ...
are
orthogonal projection In linear algebra and functional analysis Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common const ...

s. The relation between quantum mechanical symmetries and unitary operators provided an impetus for the development of the
unitary Unitary may refer to: * Unitary construction, in automotive design a common term for unibody (unitary body/chassis) construction * Lethal Unitary Chemical Agents and Munitions (Unitary), as chemical weapons opposite of Binary * Unitarianism, in Chr ...
representation theory Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...
of
groups A group is a number of people 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 identi ...
, 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 (mathematics), function describes the time dependence of a Point (geometry), point in a Manifold, geometrical space. Examples include the mathematical models that describe the ...
can be analyzed using Hilbert space techniques in the framework of
ergodic theory Ergodic theory ( Greek: ' "work", ' "way") is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
. The algebra of
observable In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through ...
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 key pioneers of quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstr ...
's
matrix mechanics#REDIRECT Matrix mechanics Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born Max Born (; 11 December 1882 – 5 January 1970) was a German physicist A physicist is a scientist A scientist ...
formulation of quantum theory. Von Neumann began investigating
operator algebra In functional analysis Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional ...
s in the 1930s, as
rings Ring most commonly refers either to a hollow circular shape or to a high-pitched sound. It thus may refer to: *Ring (jewellery), a circular, decorative or symbolic ornament worn on fingers, toes, arm or neck Ring may also refer to: Sounds * Ri ...
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 linear operator, bounded operators on a Hilbert space that is Closed set, closed in the weak operator topology and contains the identity operator. It is a special type of ...
s. In the 1940s,
Israel Gelfand Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gel'fand, or Izrail M. Gelfand ( yi, ישראל געלפֿאַנד, russian: Изра́иль Моисе́евич Гельфа́нд; – 5 October 2009) was a prominent USSR, Soviet m ...
, Mark Naimark and
Irving Segal Irving Ezra Segal (1918–1998) was an American mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory) ...
gave a definition of a kind of operator algebras called
C*-algebra In mathematics, specifically in functional analysis Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a commo ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s associated to measure spaces , where is a set, is a σ-algebra of subsets of , and is a countably additive measure on . Let be the space of those complex-valued measurable functions on for which the
Lebesgue integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
of the square of the
absolute value In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

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 File:Riemann integral irregular.gif, The partition does not need to be regular, as shown here. The approximation works as long as the width of each subdivision tends to zero. In the branch of mathematics known as real analysis, the Riemann integ ...
. The Lebesgue spaces appear in many natural settings. The spaces and of square-integrable functions with respect to the
Lebesgue measure In Measure (mathematics), measure theory, a branch of mathematics, the Lebesgue measure, named after france, French mathematician Henri Lebesgue, is the standard way of assigning a measure (mathematics), measure to subsets of ''n''-dimensional Eucli ...
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 normed space, norm that is a combination of Lp norm, ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a s ...
s, denoted by or , are Hilbert spaces. These are a special kind of
function space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
in which
differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Product differentiation, in marketing * Differentiated service, a service that varies with the identity o ...

may be performed, but that (unlike other
Banach spaces In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
such as the Hölder spaces) support the structure of an inner product. Because differentiation is permitted, Sobolev spaces are a convenient setting for the theory of
partial differential equations In , a partial differential equation (PDE) is an equation which imposes relations between the various s of a . The function is often thought of as an "unknown" to be solved for, similarly to how is thought of as an unknown number to be solved ...
. They also form the basis of the theory of direct methods in the calculus of variations. For a non-negative integer and , the Sobolev space contains functions whose
weak derivative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 potentials; roughly, $H^s(\Omega) = \left. \left\ \,.$ Here is the Laplacian and is understood in terms of the spectral mapping theorem. Apart from providing a workable definition of Sobolev spaces for non-integer , this definition also has particularly desirable properties under the
Fourier transform#REDIRECT Fourier transform In mathematics, a Fourier transform (FT) is a Integral transform, mathematical transform that decomposes function (mathematics), functions depending on space or time into functions depending on spatial or temporal frequenc ...
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 A pact, from Latin ''pactum'' ("something agreed upon"), is a formal agreement. In international relations International relations (IR), international affairs (IA) or internationa ...
Riemannian manifold In differential geometry Differential geometry is a mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a c ...
, one can obtain for instance the
Hodge decomposition In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, which is the basis of
Hodge theory In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
.

## 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 Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Der ...
and
harmonic analysis Harmonic analysis is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathe ...
, whose elements are certain
holomorphic function A rectangular grid (top) and its image under a conformal map ''f'' (bottom). In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algeb ...
s in a complex domain. Let denote the
unit disc An open Euclidean unit disk In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathemati ...

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 spaceIn complex analysis, functional analysis and operator theory, a Bergman space is a function space of holomorphic functions in a Domain (mathematical analysis), domain ''D'' of the complex plane that are sufficiently well-behaved at the boundary that ...
s are another family of Hilbert spaces of holomorphic functions. Let be a bounded open set in the
complex plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
(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 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 A pact, from Latin ''pactum'' ("something agreed upon"), is a formal agreement. In international relations International relations (IR), international affairs (IA) or internationa ...
subsets of , that $\sup_ \left, f(z)\ \le C_K \left\, f\right\, _2 \,,$ which in turn follows from
Cauchy's integral formula In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary ...
. Thus convergence of a sequence of holomorphic functions in implies also
compact convergence In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, 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 (mathematics), function from its original function space into another function space via integral, integration, where some of the properties of the original function might be more easily charac ...
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 Reproduction (or procreation or breeding) is the biological process Biological processes are those processes that are vital for an organism to live, and that shape its capacities for interacting with its environment. Biological processes a ...
, 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. Reproducing kernels are common in other areas of mathematics as well. For instance, in
harmonic analysis Harmonic analysis is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathe ...
the Poisson kernel is a reproducing kernel for the Hilbert space of square-integrable harmonic functions in the unit ball. That the latter is a Hilbert space at all is a consequence of the mean value theorem for harmonic functions.

# Applications

Many of the applications of Hilbert spaces exploit the fact that Hilbert spaces support generalizations of simple geometric concepts like projection operator, projection and change of basis from their usual finite dimensional setting. In particular, the spectral theory of continuous function, continuous self-adjoint operator, self-adjoint linear operators on a Hilbert space generalizes the usual Eigendecomposition of a matrix, spectral decomposition of a matrix (mathematics), matrix, 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 equations, spectral methods on a suitable Hilbert space are used to study the behavior of eigenvalues and eigenfunctions of differential equations. For example, the Sturm–Liouville theory, Sturm–Liouville problem arises in the study of the harmonics of waves in a violin string or a drum, and is a central problem in ordinary differential equations. The problem is a differential equation of the form $-\frac\left[p(x)\frac\right] + q(x)y = \lambda w(x)y$ 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 defined by the Green's function 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 , a partial differential equation (PDE) is an equation which imposes relations between the various s of a . The function is often thought of as an "unknown" to be solved for, similarly to how is thought of as an unknown number to be solved ...
.. For many classes of partial differential equations, such as linear elliptic partial differential equation, elliptic equations, it is possible to consider a generalized solution (known as a weak derivative, weak solution) by enlarging the class of functions. Many weak formulations involve the class of Sobolev space, Sobolev functions, which is a Hilbert space. A suitable weak formulation reduces to a geometrical problem the analytic problem of finding a solution or, often what is more important, showing that a solution exists and is unique for given boundary data. For linear elliptic equations, one geometrical result that ensures unique solvability for a large class of problems is the Lax–Milgram theorem. This strategy forms the rudiment of the Galerkin method (a finite element method) for numerical solution of partial differential equations. A typical example is the Poisson equation with Dirichlet boundary conditions 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, and is a continuous linear functional, 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 partial differential equation, elliptic, it follows from Poincaré's inequality that the bilinear form is coercive function, coercive. The Lax–Milgram theorem then ensures the existence and uniqueness of solutions of this equation. Hilbert spaces allow for many elliptic partial differential equations to be formulated in a similar way, and the Lax–Milgram theorem is then a basic tool in their analysis. With suitable modifications, similar techniques can be applied to parabolic partial differential equations and certain hyperbolic partial differential equations.

## Ergodic theory

The field of
ergodic theory Ergodic theory ( Greek: ' "work", ' "way") is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
is the study of the long-term behavior of chaos theory, chaotic dynamical systems. The protypical case of a field that ergodic theory applies to is
thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...

, in which—though the microscopic state of a system is extremely complicated (it is impossible to understand the ensemble of individual collisions between particles of matter)—the average behavior over sufficiently long time intervals is tractable. The laws of thermodynamics are assertions about such average behavior. In particular, one formulation of the zeroth law of thermodynamics asserts that over sufficiently long timescales, the only functionally independent measurement that one can make of a thermodynamic system in equilibrium is its total energy, in the form of temperature. An ergodic dynamical system is one for which, apart from the energy—measured by the Hamiltonian (quantum mechanics), Hamiltonian—there are no other functionally independent conserved quantities on the phase space. 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 (Hamiltonian), Liouville's theorem implies that there exists a measure theory, measure on the energy surface that is invariant under the time translation. As a result, time translation is a unitary transformation 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis ...
is to decompose a function into a (possibly infinite) linear combination of given basis functions: the associated Fourier series. The classical Fourier series associated to a function 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 convergence, mean). The problem can also be studied from the abstract point of view: every Hilbert space has an
orthonormal basisIn linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of u ...
, 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 or wavelets for instance, and in higher dimensions into spherical harmonics. 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, best approximation is the #Orthogonal complements and projections, orthogonal projection 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, Bessel's inequality and Parseval's formula. In various applications to physical problems, a function can be decomposed into physically meaningful eigenfunctions of a differential operator (typically the Laplace operator): this forms the foundation for the spectral study of functions, in reference to the spectral theorem, spectrum of the differential operator. A concrete physical application involves the problem of hearing the shape of a drum: given the fundamental modes of vibration that a drumhead is capable of producing, can one infer the shape of the drum itself? The mathematical formulation of this question involves the Dirichlet eigenvalues of the Laplace equation in the plane, that represent the fundamental modes of vibration in direct analogy with the integers that represent the fundamental modes of vibration of the violin string. Spectral theory also underlies certain aspects of the
Fourier transform#REDIRECT Fourier transform In mathematics, a Fourier transform (FT) is a Integral transform, mathematical transform that decomposes function (mathematics), functions depending on space or time into functions depending on spatial or temporal frequenc ...
of a function. Whereas Fourier analysis decomposes a function defined on a compact set into the discrete spectrum of the Laplacian (which corresponds to the vibrations of a violin string or drum), the Fourier transform of a function is the decomposition of a function defined on all of Euclidean space into its components in the continuous spectrum of the Laplacian. The Fourier transformation is also geometrical, in a sense made precise by the Plancherel theorem, that asserts that it is an isometry of one Hilbert space (the "time domain") with another (the "frequency domain"). This isometry property of the Fourier transformation is a recurring theme in abstract
harmonic analysis Harmonic analysis is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathe ...
(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, developed by
John von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian Americans, Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. Von Neumann was generally rega ...

, the possible states (more precisely, the pure states) of a quantum mechanical system are represented by unit vectors (called ''state vectors'') residing in a complex separable Hilbert space, known as the State space (physics), state space, well defined up to a complex number of norm 1 (the phase factor). In other words, the possible states are points in the projective space, projectivization of a Hilbert space, usually called the complex projective space. The exact nature of this Hilbert space is dependent on the system; for example, the position and momentum states for a single non-relativistic spin zero particle is the space of all
square-integrable In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
functions, while the states for the spin of a single proton are unit elements of the two-dimensional complex Hilbert space of spinors in three dimensions, spinors. Each observable is represented by a self-adjoint operator, self-adjoint linear operator acting on the state space. Each eigenstate of an observable corresponds to an eigenvector of the operator, and the associated eigenvalue corresponds to the value of the observable in that eigenstate. The inner product between two state vectors is a complex number known as a probability amplitude. During an ideal measurement of a quantum mechanical system, the probability that a system collapses from a given initial state to a particular eigenstate is given by the square of the
absolute value In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

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 matrix, density matrices: self-adjoint operators of trace of a matrix, 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 colors. As physical colors can be composed of any number of spectral colors, the space of physical colors may aptly be represented by a Hilbert space over spectral colors. Humans have Trichromacy, 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 (color), 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
. 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. 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.

## 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, Orthogonal complements and projections). This result is especially significant in applied mathematics, especially numerical analysis, where it forms the basis of least squares methods. In particular, when 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 continuous dual space, dual space is the space of all continuous function (topology), continuous 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 A parallelogram. The sides are shown in blue and the diagonals in red. In mathematics, the simplest form of the parallelogram law (also called the parallelogram identity) belongs to elementary geometry. It states that the sum of the squares of the ...

, and so the dual space is also an inner product space where this inner product can be defined in terms of this dual norm by using the polarization identity. The dual space is also complete so it is a Hilbert space in its own right. If 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 The mathematical concept of a Hilbert space, named after David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and one o ...
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 map, surjective, which makes this map an Isometry, isometric Antilinear map, antilinear 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 (algebra), kernel 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 popular in
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

. 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 Banach space, 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 space, reflexive, meaning that the natural map from into its dual space, double dual space is an isomorphism.

## Weakly-convergent sequences

In a Hilbert space , a sequence is weak topology#Weak convergence, 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, Bessel's inequality. Every weakly convergent sequence is bounded, by the uniform boundedness principle. Conversely, every bounded sequence in a Hilbert space admits weakly convergent subsequences (Alaoglu's theorem). This fact may be used to prove minimization results for continuous convex functionals, in the same way that the Bolzano–Weierstrass theorem is used for continuous functions on . 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 method in the calculus of variations, direct methods in the calculus of variations. Minimization results for convex functionals are also a direct consequence of the slightly more abstract fact that closed bounded convex subsets in a Hilbert space are Weak topology, weakly compact, since is reflexive. The existence of weakly convergent subsequences is a special case of the Eberlein–Šmulian theorem.

## Banach space properties

Any general property of
Banach space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
s continues to hold for Hilbert spaces. The open mapping theorem (functional analysis), open mapping theorem states that a continuous function, continuous surjective linear transformation from one Banach space to another is an open mapping meaning that it sends open sets to open sets. A corollary is the bounded inverse theorem, that a continuous and bijective linear function from one Banach space to another is an isomorphism (that is, a continuous linear map whose inverse is also continuous). This theorem is considerably simpler to prove in the case of Hilbert spaces than in general Banach spaces. The open mapping theorem is equivalent to the closed graph theorem, which asserts that a linear function from one Banach space to another is continuous if and only if its graph is a closed set. In the case of Hilbert spaces, this is basic in the study of unbounded operators (see closed operator). The (geometrical) Hahn–Banach theorem asserts that a closed convex set can be separated from any point outside it by means of a hyperplane of the Hilbert space. This is an immediate consequence of the #Best approximation, 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 function (topology), continuous linear operators from a Hilbert space to a second Hilbert space are Bounded linear operator, ''bounded'' in the sense that they map bounded sets to bounded sets. Conversely, if an operator is bounded, then it is continuous. The space of such bounded linear operators has a
norm Norm, the Norm or NORM may refer to: In academic disciplines * Norm (geology), an estimate of the idealised mineral content of a rock * Norm (philosophy) Norms are concepts ( sentences) of practical import, oriented to effecting an action, rat ...
, the operator norm 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 The mathematical concept of a Hilbert space, named after David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and one o ...
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 Hermitian adjoint, 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 Riesz representation theorem#Adjoints and transposes, identical to the Transpose of a linear map, transpose 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 Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a commo ...
, which is a type of
operator algebra In functional analysis Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional ...
. 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 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 of a matrix, square root such that $A = B^2 = B^*B\,.$ In a sense made precise by the #Spectral theorem, spectral theorem, 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 operators 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 operator, unitary 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 (mathematics), group under composition, which is the isometry group of . An element of is compact operator, compact if it sends bounded sets to relatively compact sets. Equivalently, a bounded operator is compact if, for any bounded sequence , the sequence has a convergent subsequence. Many integral operators are compact, and in fact define a special class of operators known as Hilbert–Schmidt operators that are especially important in the study of integral equations. Fredholm operators differ from a compact operator by a multiple of the identity, and are equivalently characterized as operators with a finite dimensional kernel (linear operator), kernel and cokernel. The index of a Fredholm operator is defined by $\operatorname T = \dim\ker T - \dim\operatorname T \,.$ The index is homotopy invariant, and plays a deep role in differential geometry via the Atiyah–Singer index theorem.

## Unbounded operators

Unbounded operators are also tractable in Hilbert spaces, and have important applications to quantum mechanics. An unbounded operator 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 operator, 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 and position operator, 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 direct sum of modules#Direct sum of Hilbert spaces, (orthogonal) direct sum, and denoted $H_1 \oplus H_2 \,,$ consisting of the set of all ordered pairs 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 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 operators that satisfy the orthogonality condition $E_i E_j = 0,\quad i \neq j \,.$ The spectral theorem for compact operator, compact 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 of a system containing a variable number of particles, where each Hilbert space in the direct sum corresponds to an additional degrees of freedom (mechanics), degree of freedom for the quantum mechanical system. In
representation theory Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...
, the Peter–Weyl theorem guarantees that any unitary representation of a compact group on a Hilbert space splits as the direct sum of finite-dimensional representations.

## Tensor products

If and , then one defines an inner product on the (ordinary) tensor product 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 Sesquilinear form, 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 operators from to .

# Orthonormal bases

The notion of an
orthonormal basisIn linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of u ...
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 of the family , , is dense set, 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 set, countable). Such a system is always linearly independent. Completeness of an orthonormal system of vectors of a Hilbert space can be equivalently restated as: : if for all and some 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'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...
; * 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 matrix (mat ...
; 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 set, closed and Bounded set, bounded is not necessarily Sequentially compact space, (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 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 series (mathematics)#Summations over arbitrary index sets, 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 identity, 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 cardinal number, cardinality, 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). 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 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 space, separable provided it contains a dense countable subset. Along with Zorn's lemma, this means a Hilbert space is separable if and only if it admits a countable orthonormal basis. All infinite-dimensional separable Hilbert spaces are therefore isometrically isomorphic to . 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, most of the Hilbert spaces are in fact separable, as stipulated by the Wightman axioms. However, it is sometimes argued that non-separable Hilbert spaces are also important in quantum field theory, roughly because the systems in the theory possess an infinite number of degrees of freedom (mechanics), degrees of freedom and any infinite Tensor product of Hilbert spaces, Hilbert tensor product (of spaces of dimension greater than one) is non-separable. For instance, a bosonic field can be naturally thought of as an element of a tensor product whose factors represent harmonic oscillators at each point of space. From this perspective, the natural state space of a boson might seem to be a non-separable space. However, it is only a small separable subspace of the full tensor product that can contain physically meaningful fields (on which the observables can be defined). Another non-separable Hilbert space models the state of an infinite collection of particles in an unbounded region of space. An orthonormal basis of the space is indexed by the density of the particles, a continuous parameter, and since the set of possible densities is uncountable, the basis is not countable.

# Orthogonal complements and projections

If is a subset of a Hilbert space , the set of vectors orthogonal to is defined by $S^\perp = \left\ \,.$ The set is a closed 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 transformation, natural 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 $i : V \to H \,,$ 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 the sense that if , then with equality holding if and only if is contained in the closure (topology), closure of . This result is a special case of the Hahn–Banach theorem. The closure of a subspace can be completely characterized in terms of the orthogonal complement: if is a subspace of , then the closure of is equal to . The orthogonal complement is thus a Galois connection on the partial order of subspaces of a Hilbert space. In general, the orthogonal complement of a sum of subspaces is the intersection of the orthogonal complements: $\left(\sum_i V_i\right)^\perp = \bigcap_i V_i^\perp \,.$ If the are in addition closed, then $\overline = \left(\bigcap_i V_i\right)^\perp \,.$

# Spectral theory

There is a well-developed spectral theory for self-adjoint operators in a Hilbert space, that is roughly analogous to the study of symmetric matrix, 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 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 of a self-adjoint operator takes a particularly simple form if, in addition, is assumed to be a compact operator. The Compact operator on Hilbert space#Spectral theorem, 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 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 operators. The general spectral theorem for self-adjoint operators involves a kind of operator-valued Riemann–Stieltjes integral, rather than an infinite summation. The ''spectral family'' associated to 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. A major application of spectral methods is the spectral mapping theorem, 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 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. 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

Thomas Pynchon introduced the fictional character, Sammy Hilbert-Spaess (a pun on "Hilbert Space"), in his 1973 novel, Gravity's Rainbow. Hilbert-Spaess is first described as a "a ubiquitous double agent" and later as "at least a double agent". The novel had earlier referenced the work of fellow German mathematician Kurt Gödel's Gödel's incompleteness theorems, Incompleteness Theorems, which showed that Hilbert's program, Hilbert's Program, Hilbert's formalized plan to unify mathematics into a single set of axioms, was not possible.

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# References

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