
In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a Hilbert space is a
real or
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
inner product space
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, ofte ...
that is also a
complete metric space
In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in .
Intuitively, a space is complete if there are no "points missing" from it (inside or at the bou ...
with respect to the metric induced by the inner product. It generalizes the notion of
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
. The
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, ofte ...
allows lengths and angles to be defined. Furthermore,
completeness means that there are enough
limits
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2009 ...
in the space to allow the techniques of calculus to be used. A Hilbert space is a special case of a
Banach space
In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
.
Hilbert spaces were studied beginning in the first decade of the 20th century by
David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of his time.
Hilbert discovered and developed a broad range of fundamental idea ...
,
Erhard Schmidt, and
Frigyes Riesz
Frigyes Riesz (, , sometimes known in English and French as Frederic Riesz; 22 January 1880 – 28 February 1956) was a HungarianEberhard Zeidler: Nonlinear Functional Analysis and Its Applications: Linear monotone operators. Springer, 199/ref> ...
. They are indispensable tools in the theories of
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives.
The function is often thought of as an "unknown" that solves the equation, similar to ho ...
s,
quantum mechanics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
,
Fourier analysis
In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fo ...
(which includes applications to
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, Scalar potential, potential fields, Seismic tomograph ...
and
heat transfer
Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction, ...
), and
ergodic theory
Ergodic theory is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, "statistical properties" refers to properties which are expressed through the behav ...
(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 ( ; ; December 28, 1903 – February 8, 1957) was a Hungarian and American mathematician, physicist, computer scientist and engineer. Von Neumann had perhaps the widest coverage of any mathematician of his time, in ...
coined the term ''Hilbert space'' for the abstract concept that underlies many of these diverse applications. The success of Hilbert space methods ushered in a very fruitful era for
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...
. Apart from the classical Euclidean vector spaces, examples of Hilbert spaces include
spaces of square-integrable functions,
spaces of sequences,
Sobolev space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...
s consisting of
generalized function
In mathematics, generalized functions are objects extending the notion of functions on real or complex numbers. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful for tr ...
s, and
Hardy space
In complex analysis, the Hardy spaces (or Hardy classes) H^p are 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 . In real anal ...
s of
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...
s.
Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the
Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
and
parallelogram law
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 lengths of the four sides of a parallelogram equals the sum of the s ...
hold in a Hilbert space. At a deeper level,
perpendicular
In geometry, two geometric objects are perpendicular if they intersect at right angles, i.e. at an angle of 90 degrees or π/2 radians. The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', � ...
projection
Projection or projections may refer to:
Physics
* Projection (physics), the action/process of light, heat, or sound reflecting from a surface to another in a different direction
* The display of images by a projector
Optics, graphics, and carto ...
onto a
linear subspace
In mathematics, the term ''linear'' is used in two distinct senses for two different properties:
* linearity of a ''function (mathematics), function'' (or ''mapping (mathematics), mapping'');
* linearity of a ''polynomial''.
An example of a li ...
plays a significant role in
optimization
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives. It is generally divided into two subfiel ...
problems and other aspects of the theory. An element of a Hilbert space can be uniquely specified by its coordinates with respect to an
orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite Dimension (linear algebra), dimension is a Basis (linear algebra), basis for V whose vectors are orthonormal, that is, they are all unit vec ...
, in analogy with
Cartesian coordinates
In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...
in classical geometry. When this
basis is
countably infinite
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
, it allows identifying the Hilbert space with the space of the
infinite sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is call ...
s that are
square-summable. The latter space is often in the older literature referred to as ''the'' Hilbert space.
Definition and illustration
Motivating example: Euclidean vector space
One of the most familiar examples of a Hilbert space is the
Euclidean vector space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces'' ...
consisting of three-dimensional
vectors, denoted by , and equipped with the
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a Scalar (mathematics), scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. N ...
. The dot product takes two vectors and , and produces a real number . If and are represented in
Cartesian coordinates
In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...
, then the dot product is defined by
The dot product satisfies the properties
# It is
symmetric
Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
in and : .
# It is
linear
In mathematics, the term ''linear'' is used in two distinct senses for two different properties:
* linearity of a '' function'' (or '' mapping'');
* linearity of a '' polynomial''.
An example of a linear function is the function defined by f(x) ...
in its first argument: for any
scalars
Scalar may refer to:
*Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
*Scalar (physics), a physical quantity that can be described by a single element of a number field such a ...
, , and vectors , , and .
# It is
positive definite: for all vectors , , with equality
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
.
An operation on pairs of vectors that, like the dot product, satisfies these three properties is known as a (real)
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, ofte ...
. A
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
equipped with such an inner product is known as a (real)
inner product space
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, ofte ...
. Every finite-dimensional inner product space is also a Hilbert space. The basic feature of the dot product that connects it with Euclidean geometry is that it is related to both the length (or
norm) of a vector, denoted , and to the angle between two vectors and by means of the formula
Multivariable calculus
Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving multiple variables ('' mult ...
in Euclidean space relies on the ability to compute
limits
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2009 ...
, and to have useful criteria for concluding that limits exist. A
mathematical series
consisting of vectors in is
absolutely convergent
In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is said ...
provided that the sum of the lengths converges as an ordinary series of real numbers:
Just as with a series of scalars, a series of vectors that converges absolutely also converges to some limit vector in the Euclidean space, in the sense that
This property expresses the ''completeness'' of Euclidean space: that a series that converges absolutely also converges in the ordinary sense.
Hilbert spaces are often taken over the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s. The
complex plane
In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
denoted by is equipped with a notion of magnitude, the
complex modulus , which is defined as the square root of the product of with its
complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a and b are real numbers, then the complex conjugate of a + bi is a - ...
:
If is a decomposition of into its real and imaginary parts, then the modulus is the usual Euclidean two-dimensional length:
The inner product of a pair of complex numbers and is the product of with the complex conjugate of :
This is complex-valued. The real part of gives the usual two-dimensional Euclidean
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a Scalar (mathematics), scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. N ...
.
A second example is the space whose elements are pairs of complex numbers . Then an inner product of with another such vector is given by
The real part of is then the four-dimensional Euclidean dot product. This inner product is ''Hermitian'' symmetric, which means that the result of interchanging and is the complex conjugate:
Definition
A is a
real or
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
inner product space
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, ofte ...
that is also a
complete metric space
In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in .
Intuitively, a space is complete if there are no "points missing" from it (inside or at the bou ...
with respect to the distance function
induced by the inner product.
[The mathematical material in this section can be found in any good textbook on functional analysis, such as , , or .]
To say that a complex vector space is a means that there is an inner product
associating a complex number to each pair of elements
of that satisfies the following properties:
# The inner product is conjugate symmetric; that is, the inner product of a pair of elements is equal to the
complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a and b are real numbers, then the complex conjugate of a + bi is a - ...
of the inner product of the swapped elements:
Importantly, this implies that
is a real number.
# The inner product is
linear
In mathematics, the term ''linear'' is used in two distinct senses for two different properties:
* linearity of a '' function'' (or '' mapping'');
* linearity of a '' polynomial''.
An example of a linear function is the function defined by f(x) ...
in its first
[In some conventions, inner products are linear in their second arguments instead.] argument. For all complex numbers
and
# The inner product of an element with itself is
positive definite:
It follows from properties 1 and 2 that a complex inner product is , also called , in its second argument, meaning that
A is defined in the same way, except that is a real vector space and the inner product takes real values. Such an inner product will be a
bilinear map
In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example.
A bilinear map can also be defined for ...
and
will form a
dual system
In mathematics, a dual system, dual pair or a duality over a field \mathbb is a triple (X, Y, b) consisting of two vector spaces, X and Y, over \mathbb and a non- degenerate bilinear map b : X \times Y \to \mathbb.
In mathematics, duality is t ...
.
The
norm is the real-valued function
and the distance
between two points
in is defined in terms of the norm by
That this function is a distance function means firstly that it is symmetric in
and
secondly that the distance between
and itself is zero, and otherwise the distance between
and
must be positive, and lastly that the
triangle inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
This statement permits the inclusion of Degeneracy (mathematics)#T ...
holds, meaning that the length of one leg of a triangle cannot exceed the sum of the lengths of the other two legs:
:

This last property is ultimately a consequence of the more fundamental
Cauchy–Schwarz inequality
The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is an upper bound on the absolute value of the inner product between two vectors in an inner product space in terms of the product of the vector norms. It is ...
, which asserts
with equality if and only if
and
are
linearly dependent
In the theory of vector spaces, a set of vectors is said to be if there exists no nontrivial linear combination of the vectors that equals the zero vector. If such a linear combination exists, then the vectors are said to be . These concepts ...
.
With a distance function defined in this way, any inner product space is a
metric space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
, and sometimes is known as a . Any pre-Hilbert space that is additionally also a
complete space
In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in .
Intuitively, a space is complete if there are no "points missing" from it (inside or at the bo ...
is a Hilbert space.
The of is expressed using a form of the
Cauchy criterion
The Cauchy convergence test is a method used to test infinite series for convergence. It relies on bounding sums of terms in the series. This convergence criterion is named after Augustin-Louis Cauchy who published it in his textbook '' Cours d'A ...
for sequences in : a pre-Hilbert space is complete if every
Cauchy sequence
In mathematics, a Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all excluding a finite number of elements of the sequence are le ...
converges with respect to this norm to an element in the space. Completeness can be characterized by the following equivalent condition: if a series of vectors
converges absolutely in the sense that
then the series converges in , in the sense that the
partial sums converge to an element of .
As a complete normed space, Hilbert spaces are by definition also
Banach space
In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
s. As such they are
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
s, in which
topological
Topology (from the Greek words , and ) is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, wit ...
notions like the
openness
Openness is an overarching concept that is characterized by an emphasis on transparency and collaboration. That is, openness refers to "accessibility of knowledge, technology and other resources; the transparency of action; the permeability of or ...
and
closedness of subsets are
well defined. Of special importance is the notion of a closed
linear subspace
In mathematics, the term ''linear'' is used in two distinct senses for two different properties:
* linearity of a ''function (mathematics), function'' (or ''mapping (mathematics), mapping'');
* linearity of a ''polynomial''.
An example of a li ...
of a Hilbert space that, with the inner product induced by
restriction, is also complete (being a closed set in a complete metric space) and therefore a Hilbert space in its own right.
Second example: sequence spaces
The
sequence space
In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural num ...
consists of all
infinite sequences of complex numbers such that the following series
converges:
The inner product on is defined by:
This second series converges as a consequence of the
Cauchy–Schwarz inequality
The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is an upper bound on the absolute value of the inner product between two vectors in an inner product space in terms of the product of the vector norms. It is ...
and the convergence of the previous series.
Completeness of the space holds provided that whenever a series of elements from converges absolutely (in norm), then it converges to an element of . The proof is basic in
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...
, and permits mathematical series of elements of the space to be manipulated with the same ease as series of complex numbers (or vectors in a finite-dimensional Euclidean space).
History

Prior to the development of Hilbert spaces, other generalizations of Euclidean spaces were known to
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
s and
physicist
A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe. Physicists generally are interested in the root or ultimate cau ...
s. In particular, the idea of an
abstract linear space (vector space) had gained some traction towards the end of the 19th century: this is a space whose elements can be added together and multiplied by scalars (such as
real or
complex numbers
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
) 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, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is call ...
(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 and philosopher of mathematics and one of the most influential mathematicians of his time.
Hilbert discovered and developed a broad range of fundamental idea ...
and
Erhard Schmidt's study of
integral equations
In mathematical analysis, integral equations are equations in which an unknown Function (mathematics), function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: f(x_1,x_2,x_3 ...
, that two
square-integrable real-valued functions and on an interval have an ''inner product''
:
that has many of the familiar properties of the Euclidean dot product. In particular, the idea of an
orthogonal
In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ...
family of functions has meaning. Schmidt exploited the similarity of this inner product with the usual dot product to prove an analog of the
spectral decomposition for an operator of the form
:
where is a continuous function symmetric in and . The resulting
eigenfunction expansion expresses the function as a series of the form
:
where the functions are orthogonal in the sense that for all . The individual terms in this series are sometimes referred to as elementary product solutions. However, there are eigenfunction expansions that fail to converge in a suitable sense to a square-integrable function: the missing ingredient, which ensures convergence, is completeness.
The second development was the
Lebesgue integral
In mathematics, the integral of a non-negative Function (mathematics), function of a single variable can be regarded, in the simplest case, as the area between the Graph of a function, graph of that function and the axis. The Lebesgue integral, ...
, an alternative to the
Riemann integral
In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Gö ...
introduced by
Henri Lebesgue
Henri Léon Lebesgue (; ; June 28, 1875 – July 26, 1941) was a French mathematician known for his Lebesgue integration, theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an ...
in 1904. The Lebesgue integral made it possible to integrate a much broader class of functions. In 1907,
Frigyes Riesz
Frigyes Riesz (, , sometimes known in English and French as Frederic Riesz; 22 January 1880 – 28 February 1956) was a HungarianEberhard Zeidler: Nonlinear Functional Analysis and Its Applications: Linear monotone operators. Springer, 199/ref> ...
and
Ernst Sigismund Fischer independently proved that the space of square Lebesgue-integrable functions is a
complete metric space
In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in .
Intuitively, a space is complete if there are no "points missing" from it (inside or at the bou ...
. As a consequence of the interplay between geometry and completeness, the 19th century results of
Joseph Fourier
Jean-Baptiste Joseph Fourier (; ; 21 March 1768 – 16 May 1830) was a French mathematician and physicist born in Auxerre, Burgundy and best known for initiating the investigation of Fourier series, which eventually developed into Fourier analys ...
,
Friedrich Bessel
Friedrich Wilhelm Bessel (; 22 July 1784 – 17 March 1846) was a German astronomer, mathematician, physicist, and geodesy, geodesist. He was the first astronomer who determined reliable values for the distance from the Sun to another star by th ...
and
Marc-Antoine Parseval on
trigonometric series
In mathematics, trigonometric series are a special class of orthogonal series of the form
: A_0 + \sum_^\infty A_n \cos + B_n \sin,
where x is the variable and \ and \ are coefficients. It is an infinite version of a trigonometric polynom ...
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
The Riesz representation theorem, sometimes called the Riesz–Fréchet representation theorem after Frigyes Riesz and Maurice René Fréchet, establishes an important connection between a Hilbert space and its continuous dual space. If the un ...
was independently established by
Maurice Fréchet
Maurice may refer to:
*Maurice (name), a given name and surname, including a list of people with the name
Places
* or Mauritius, an island country in the Indian Ocean
* Maurice, Iowa, a city
* Maurice, Louisiana, a village
* Maurice River, a t ...
and
Frigyes Riesz
Frigyes Riesz (, , sometimes known in English and French as Frederic Riesz; 22 January 1880 – 28 February 1956) was a HungarianEberhard Zeidler: Nonlinear Functional Analysis and Its Applications: Linear monotone operators. Springer, 199/ref> ...
in 1907.
John von Neumann
John von Neumann ( ; ; December 28, 1903 – February 8, 1957) was a Hungarian and American mathematician, physicist, computer scientist and engineer. Von Neumann had perhaps the widest coverage of any mathematician of his time, in ...
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 mathematician, theoretical physicist, logician and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, ...
and
Norbert Wiener
Norbert Wiener (November 26, 1894 – March 18, 1964) was an American computer scientist, mathematician, and philosopher. He became a professor of mathematics at the Massachusetts Institute of Technology ( MIT). A child prodigy, Wiener late ...
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 Wigner (, ; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist who also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his contributions to the theory of th ...
. The name "Hilbert space" was soon adopted by others, for example by Hermann Weyl in his book on quantum mechanics and the theory of groups.
The significance of the concept of a Hilbert space was underlined with the realization that it offers one of the best
mathematical formulations of quantum mechanics. In short, the states of a quantum mechanical system are vectors in a certain Hilbert space, the observables are
hermitian operator
In mathematics, a self-adjoint operator on a complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle is a linear map ''A'' (from ''V'' to itself) that is its own adjoint. That is, \langle Ax,y \rangle = \langle x,Ay \rangle for al ...
s on that space, the
symmetries
Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
of the system are
unitary operator
In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product.
Non-trivial examples include rotations, reflections, and the Fourier operator.
Unitary operators generalize unitar ...
s, and
measurements
Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events.
In other words, measurement is a process of determining how large or small a physical quantity is as compared to ...
are
orthogonal projection
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it we ...
s. The relation between quantum mechanical symmetries and unitary operators provided an impetus for the development of the
unitary
Unitary may refer to:
Mathematics
* Unitary divisor
* Unitary element
* Unitary group
* Unitary matrix
* Unitary morphism
* Unitary operator
* Unitary transformation
* Unitary representation
* Unitarity (physics)
* ''E''-unitary inverse semigr ...
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, 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 an ambient space, such as in a parametric curve. Examples include the mathematical models ...
can be analyzed using Hilbert space techniques in the framework of
ergodic theory
Ergodic theory is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, "statistical properties" refers to properties which are expressed through the behav ...
.
The algebra of
observable
In physics, an observable is a physical property or physical quantity that can be measured. In classical mechanics, an observable is a real-valued "function" on the set of all possible system states, e.g., position and momentum. In quantum ...
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, one of the main pioneers of the theory of quantum mechanics and a principal scientist in the German nuclear program during World War II.
He pub ...
's
matrix mechanics
Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. It was the first conceptually autonomous and logically consistent formulation of quantum mechanics. Its account of quantum ...
formulation of quantum theory. Von Neumann began investigating
operator algebra
In functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with the multiplication given by the composition of mappings.
The results obtained in the study o ...
s in the 1930s, as
rings of operators on a Hilbert space. The kind of algebras studied by von Neumann and his contemporaries are now known as
von Neumann algebra
In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra.
Von Neumann al ...
s. In the 1940s,
Israel Gelfand
Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gel'fand, or Izrail M. Gelfand (, , ; – 5 October 2009) was a prominent Soviet and American mathematician, one of the greatest mathematicians of the 20th century, biologist, teache ...
,
Mark Naimark and
Irving Segal
Irving Ezra Segal (1918–1998) was an American mathematician known for work on theoretical quantum mechanics. He shares credit for what is often referred to as the Segal–Shale–Weil representation. Early in his career Segal became known for h ...
gave a definition of a kind of operator algebras called
C*-algebra
In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of contin ...
s that on the one hand made no reference to an underlying Hilbert space, and on the other extrapolated many of the useful features of the operator algebras that had previously been studied. The spectral theorem for self-adjoint operators in particular that underlies much of the existing Hilbert space theory was generalized to C*-algebras. These techniques are now basic in abstract harmonic analysis and representation theory.
Examples
Lebesgue spaces
Lebesgue spaces are
function space
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a ve ...
s associated to
measure spaces , where is a set, is a
σ-algebra
In mathematical analysis and in probability theory, a σ-algebra ("sigma algebra") is part of the formalism for defining sets that can be measured. In calculus and analysis, for example, σ-algebras are used to define the concept of sets with a ...
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, the integral of a non-negative Function (mathematics), function of a single variable can be regarded, in the simplest case, as the area between the Graph of a function, graph of that function and the axis. The Lebesgue integral, ...
of the square of the
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
of the function is finite, i.e., for a function in ,
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
or
where the second form (conjugation of the first element) is commonly found in the
theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental p ...
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
In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Gö ...
.
The Lebesgue spaces appear in many natural settings. The spaces and of square-integrable functions with respect to the
Lebesgue measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...
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
is called the
weighted space , and is called the weight function. The inner product is defined by
The weighted space is identical with the Hilbert space where the measure of a Lebesgue-measurable set is defined by
Weighted spaces like this are frequently used to study
orthogonal polynomials
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal
In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geom ...
, because different families of orthogonal polynomials are orthogonal with respect to different weighting functions.
Sobolev spaces
Sobolev space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...
s, denoted by or , are Hilbert spaces. These are a special kind of
function space
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a ve ...
in which
differentiation may be performed, but that (unlike other
Banach spaces
In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
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 mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives.
The function is often thought of as an "unknown" that solves the equation, similar to how ...
.
They also form the basis of the theory of
direct methods in the calculus of variations.
For a non-negative
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
and , the Sobolev space contains functions whose
weak derivative
In mathematics, a weak derivative is a generalization of the concept of the derivative of a function (''strong derivative'') for functions not assumed differentiable, but only integrable, i.e., to lie in the L''p'' space L^1( ,b.
The method o ...
s of order up to are also . The inner product in is
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,
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
In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
that make it ideal for the study of
pseudodifferential operators. Using these methods on a
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
Riemannian manifold
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
, one can obtain for instance the
Hodge decomposition
In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every cohom ...
, which is the basis of
Hodge theory
In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every coho ...
.
Spaces of holomorphic functions
Hardy spaces
The
Hardy space
In complex analysis, the Hardy spaces (or Hardy classes) H^p are 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 . In real anal ...
s are function spaces, arising in
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
and
harmonic analysis
Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency. The frequency representation is found by using the Fourier transform for functions on unbounded do ...
, whose elements are certain
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...
s in a complex domain. Let denote the
unit disc
In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1:
:D_1(P) = \.\,
The closed unit disk around ''P'' is the set of points whose d ...
in the complex plane. Then the Hardy space is defined as the space of holomorphic functions on such that the means
remain bounded for . The norm on this Hardy space is defined by
Hardy spaces in the disc are related to Fourier series. A function is in if and only if
where
Thus consists of those functions that are ''L''
2 on the circle, and whose negative frequency Fourier coefficients vanish.
Bergman spaces
The
Bergman spaces are another family of Hilbert spaces of holomorphic functions. Let be a bounded open set in the
complex plane
In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
(or a higher-dimensional complex space) and let be the space of holomorphic functions in that are also in in the sense that
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 or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
subsets of , that
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 o ...
. Thus convergence of a sequence of holomorphic functions in implies also
compact convergence
In mathematics compact convergence (or uniform convergence on compact sets) is a type of convergence that generalizes the idea of uniform convergence. It is associated with the compact-open topology.
Definition
Let (X, \mathcal) be a topological ...
, and so the limit function is also holomorphic. Another consequence of this inequality is that the linear functional that evaluates a function at a point of is actually continuous on . The Riesz representation theorem implies that the evaluation functional can be represented as an element of . Thus, for every , there is a function such that
for all . The integrand
is known as the
Bergman kernel of . This
integral kernel
In mathematics, an integral transform is a type of transform (mathematics), transform that maps a function (mathematics), function from its original function space into another function space via integral, integration, where some of the propert ...
satisfies a reproducing property
A Bergman space is an example of a
reproducing kernel Hilbert space
In functional analysis, a reproducing kernel Hilbert space (RKHS) is a Hilbert space of functions in which point evaluation is a continuous linear functional. Specifically, a Hilbert space H of functions from a set X (to \mathbb or \mathbb) is ...
, 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 concerned with investigating the connections between a function and its representation in frequency. The frequency representation is found by using the Fourier transform for functions on unbounded do ...
the
Poisson kernel
In mathematics, and specifically in potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disk. The kernel can be understood as the deriv ...
is a reproducing kernel for the Hilbert space of square-integrable
harmonic function
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f\colon U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that i ...
s in the
unit ball
Unit may refer to:
General measurement
* Unit of measurement, a definite magnitude of a physical quantity, defined and adopted by convention or by law
**International System of Units (SI), modern form of the metric system
**English units, histo ...
. 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
Projection or projections may refer to:
Physics
* Projection (physics), the action/process of light, heat, or sound reflecting from a surface to another in a different direction
* The display of images by a projector
Optics, graphics, and carto ...
and
change of basis
In mathematics, an ordered basis of a vector space of finite dimension allows representing uniquely any element of the vector space by a coordinate vector, which is a sequence of scalars called coordinates. If two different bases are conside ...
from their usual finite dimensional setting. In particular, the
spectral theory
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operator (mathematics), operators in a variety of mathematical ...
of
continuous self-adjoint
In mathematics, an element of a *-algebra is called self-adjoint if it is the same as its adjoint (i.e. a = a^*).
Definition
Let \mathcal be a *-algebra. An element a \in \mathcal is called self-adjoint if
The set of self-adjoint elements ...
linear operator
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
s on a Hilbert space generalizes the usual
spectral decomposition of a
matrix
Matrix (: matrices or matrixes) or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics), a rectangular array of numbers, symbols or expressions
* Matrix (logic), part of a formula in prenex normal form
* Matrix (biology), the m ...
, and this often plays a major role in applications of the theory to other areas of mathematics and physics.
Sturm–Liouville theory

In the theory of
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematic ...
s, spectral methods on a suitable Hilbert space are used to study the behavior of eigenvalues and eigenfunctions of differential equations. For example, the
Sturm–Liouville problem arises in the study of the harmonics of waves in a violin string or a drum, and is a central problem in
ordinary differential equations
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with any other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives ...
. The problem is a differential equation of the form
for an unknown function on an interval , satisfying general homogeneous
Robin boundary conditions
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 operator
In functional analysis, a branch of mathematics, a compact operator is a linear operator T: X \to Y, where X,Y are normed vector spaces, with the property that T maps bounded subsets of X to relatively compact subsets of Y (subsets with compact ...
s applied to the
integral operator
An integral operator is an operator that involves integration. Special instances are:
* The operator of integration itself, denoted by the integral symbol
* Integral linear operators, which are linear operators induced by bilinear forms involvi ...
defined by the
Green's function
In mathematics, a Green's function (or Green function) is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.
This means that if L is a linear dif ...
for the system. Furthermore, another consequence of this general result is that the eigenvalues of the system can be arranged in an increasing sequence tending to infinity.
[The eigenvalues of the Fredholm kernel are , which tend to zero.]
Partial differential equations
Hilbert spaces form a basic tool in the study of
partial differential equations
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives.
The function is often thought of as an "unknown" that solves the equation, similar to how ...
.
For many classes of partial differential equations, such as linear
elliptic equations, it is possible to consider a generalized solution (known as a
weak
Weak may refer to:
Songs
* Weak (AJR song), "Weak" (AJR song), 2016
* Weak (Melanie C song), "Weak" (Melanie C song), 2011
* Weak (SWV song), "Weak" (SWV song), 1993
* Weak (Skunk Anansie song), "Weak" (Skunk Anansie song), 1995
* "Weak", a son ...
solution) by enlarging the class of functions. Many weak formulations involve the class of
Sobolev functions, which is a Hilbert space. A suitable weak formulation reduces to a geometrical problem, the analytic problem of finding a solution or, often what is more important, showing that a solution exists and is unique for given boundary data. For linear elliptic equations, one geometrical result that ensures unique solvability for a large class of problems is the
Lax–Milgram theorem. This strategy forms the rudiment of the
Galerkin method
In mathematics, in the area of numerical analysis, Galerkin methods are a family of methods for converting a continuous operator problem, such as a differential equation, commonly in a weak formulation, to a discrete problem by applying linear c ...
(a
finite element method
Finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat tran ...
) for numerical solution of partial differential equations.
A typical example is the
Poisson equation
Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with th ...
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:
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
where is a continuous
bilinear form
In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called '' scalars''). In other words, a bilinear form is a function that is linea ...
, and is a continuous
linear functional
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear mapIn some texts the roles are reversed and vectors are defined as linear maps from covectors to scalars from a vector space to its field of ...
, given respectively by
Since the Poisson equation is
elliptic, it follows from Poincaré's inequality that the bilinear form is
coercive
Coercion involves compelling a party to act in an involuntary manner through the use of threats, including threats to use force against that party. It involves a set of forceful actions which violate the free will of an individual in order to in ...
. The Lax–Milgram theorem then ensures the existence and uniqueness of solutions of this equation.
Hilbert spaces allow for many elliptic partial differential equations to be formulated in a similar way, and the Lax–Milgram theorem is then a basic tool in their analysis. With suitable modifications, similar techniques can be applied to
parabolic partial differential equation
A parabolic partial differential equation is a type of partial differential equation (PDE). Parabolic PDEs are used to describe a wide variety of time-dependent phenomena in, for example, engineering science, quantum mechanics and financial ma ...
s and certain
hyperbolic partial differential equation
In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first n - 1 derivatives. More precisely, the Cauchy problem can ...
s.
Ergodic theory

The field of
ergodic theory
Ergodic theory is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, "statistical properties" refers to properties which are expressed through the behav ...
is the study of the long-term behavior of
chaotic dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...
s. The protypical case of a field that ergodic theory applies to is
thermodynamics
Thermodynamics is a branch of physics that deals with heat, Work (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
The laws of thermodynamics are a set of scientific laws which define a group of physical quantities, such as temperature, energy, and entropy, that characterize thermodynamic systems in thermodynamic equilibrium. The laws also use various param ...
are assertions about such average behavior. In particular, one formulation of the
zeroth law of thermodynamics
The zeroth law of thermodynamics is one of the four principal laws of thermodynamics. It provides an independent definition of temperature without reference to entropy, which is defined in the second law. The law was established by Ralph H. Fowl ...
asserts that over sufficiently long timescales, the only functionally independent measurement that one can make of a thermodynamic system in equilibrium is its total energy, in the form of
temperature
Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...
.
An ergodic dynamical system is one for which, apart from the energy—measured by the
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
—there are no other functionally independent
conserved quantities on the
phase space
The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible state corresponds uniquely to a point in the phase space. For mechanical systems, the p ...
. 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 every invariant measurable functions on is constant
almost everywhere
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
. An invariant function is one for which
for all on and all time .
Liouville's theorem implies that there exists a
measure on the energy surface that is invariant under the
time translation
Time-translation symmetry or temporal translation symmetry (TTS) is a mathematical transformation in physics that moves the times of events through a common interval. Time-translation symmetry is the law that the laws of physics are unchanged ...
. As a result, time translation is a
unitary transformation
In mathematics, a unitary transformation is a linear isomorphism that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation.
Formal definition
More precise ...
of the Hilbert space consisting of square-integrable functions on the energy surface with respect to the inner product
The von Neumann mean ergodic theorem
states the following:
* If is a (strongly continuous) one-parameter
semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.
The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily th ...
of unitary operators on a Hilbert space , and is the orthogonal projection onto the space of common fixed points of , , then
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 ,
That is, the long time average of an observable is equal to its expectation value over an energy surface.
Fourier analysis

One of the basic goals of
Fourier analysis
In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fo ...
is to decompose a function into a (possibly infinite)
linear combination
In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...
of given basis functions: the associated
Fourier series
A Fourier series () is an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems ...
. The classical Fourier series associated to a function defined on the interval is a series of the form
where
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).
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
and, moreover, this series converges in the Hilbert space sense (that is, in the
mean).
The problem can also be studied from the abstract point of view: every Hilbert space has an
orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite Dimension (linear algebra), dimension is a Basis (linear algebra), basis for V whose vectors are orthonormal, that is, they are all unit vec ...
, and every element of the Hilbert space can be written in a unique way as a sum of multiples of these basis elements. The coefficients appearing on these basis elements are sometimes known abstractly as the Fourier coefficients of the element of the space. The abstraction is especially useful when it is more natural to use different basis functions for a space such as . In many circumstances, it is desirable not to decompose a function into trigonometric functions, but rather into
orthogonal polynomials
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal
In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geom ...
or
wavelet
A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the n ...
s for instance, and in higher dimensions into
spherical harmonics
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. The table of spherical harmonics co ...
.
For instance, if are any orthonormal basis functions of , then a given function in can be approximated as a finite linear combination
The coefficients are selected to make the magnitude of the difference as small as possible. Geometrically, the
best approximation is the
orthogonal projection
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it we ...
of onto the subspace consisting of all linear combinations of the , and can be calculated by
That this formula minimizes the difference is a consequence of
Bessel's inequality and Parseval's formula.
In various applications to physical problems, a function can be decomposed into physically meaningful
eigenfunction
In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
s of a
differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...
(typically the
Laplace operator
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a Scalar field, scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \ ...
): this forms the foundation for the spectral study of functions, in reference to the
spectrum
A spectrum (: spectra or spectrums) is a set of related ideas, objects, or properties whose features overlap such that they blend to form a continuum. The word ''spectrum'' was first used scientifically in optics to describe the rainbow of co ...
of the differential operator. A concrete physical application involves the problem of
hearing the shape of a drum
In theoretical mathematics, the conceptual problem of "hearing the shape of a drum" refers to the prospect of inferring information about the shape of a hypothetical idealized drumhead from the sound it makes when struck, i.e. from analysis of ...
: 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
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operator (mathematics), operators in a variety of mathematical ...
also underlies certain aspects of the
Fourier transform
In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
of a function. Whereas Fourier analysis decomposes a function defined on a
compact set
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., i ...
into the discrete spectrum of the Laplacian (which corresponds to the vibrations of a violin string or drum), the Fourier transform of a function is the decomposition of a function defined on all of Euclidean space into its components in the
continuous spectrum
In the physical sciences, the term ''spectrum'' was introduced first into optics by Isaac Newton in the 17th century, referring to the range of colors observed when white light was dispersion (optics), dispersed through a prism (optics), prism. ...
of the Laplacian. The Fourier transformation is also geometrical, in a sense made precise by the
Plancherel theorem
In mathematics, the Plancherel theorem (sometimes called the Parseval–Plancherel identity) is a result in harmonic analysis, proven by Michel Plancherel in 1910. It is a generalization of Parseval's theorem; often used in the fields of science ...
, that asserts that it is an
isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' me ...
of one Hilbert space (the "time domain") with another (the "frequency domain"). This isometry property of the Fourier transformation is a recurring theme in abstract
harmonic analysis
Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency. The frequency representation is found by using the Fourier transform for functions on unbounded do ...
(since it reflects the conservation of energy for the continuous Fourier Transform), as evidenced for instance by the
Plancherel theorem for spherical functions occurring in
noncommutative harmonic analysis.
Quantum mechanics

In the mathematically rigorous formulation of
quantum mechanics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
, developed by
John von Neumann
John von Neumann ( ; ; December 28, 1903 – February 8, 1957) was a Hungarian and American mathematician, physicist, computer scientist and engineer. Von Neumann had perhaps the widest coverage of any mathematician of his time, in ...
, the possible states (more precisely, the
pure state
In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a prediction for the system re ...
s) of a quantum mechanical system are represented by
unit vector
In mathematics, a unit vector in a normed vector space is a Vector (mathematics and physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...
s (called ''state vectors'') residing in a complex separable Hilbert space, known as the
state space
In computer science, a state space is a discrete space representing the set of all possible configurations of a system. It is a useful abstraction for reasoning about the behavior of a given system and is widely used in the fields of artificial ...
, well defined up to a complex number of norm 1 (the
phase factor). In other words, the possible states are points in the
projectivization
In mathematics, projectivization is a procedure which associates with a non-zero vector space a projective space , whose elements are one-dimensional subspaces of . More generally, any subset of closed under scalar multiplication defines a s ...
of a Hilbert space, usually called the
complex projective space
In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a ...
. The exact nature of this Hilbert space is dependent on the system; for example, the position and momentum states for a single non-relativistic spin zero particle is the space of all
square-integrable functions, while the states for the spin of a single proton are unit elements of the two-dimensional complex Hilbert space of
spinors
In geometry and physics, spinors (pronounced "spinner" IPA ) are elements of a complex numbers, complex vector space that can be associated with Euclidean space. A spinor transforms linearly when the Euclidean space is subjected to a slight (infi ...
. Each observable is represented by a
self-adjoint
In mathematics, an element of a *-algebra is called self-adjoint if it is the same as its adjoint (i.e. a = a^*).
Definition
Let \mathcal be a *-algebra. An element a \in \mathcal is called self-adjoint if
The set of self-adjoint elements ...
linear operator
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
acting on the state space. Each eigenstate of an observable corresponds to an
eigenvector
In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by ...
of the operator, and the associated
eigenvalue
In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
corresponds to the value of the observable in that eigenstate.
The inner product between two state vectors is a complex number known as a
probability amplitude
In quantum mechanics, a probability amplitude is a complex number used for describing the behaviour of systems. The square of the modulus of this quantity at a point in space represents a probability density at that point.
Probability amplitu ...
. During an ideal measurement of a quantum mechanical system, the probability that a system collapses from a given initial state to a particular eigenstate is given by the square of the
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
of the probability amplitudes between the initial and final states. The possible results of a measurement are the eigenvalues of the operator—which explains the choice of self-adjoint operators, for all the eigenvalues must be real. The probability distribution of an observable in a given state can be found by computing the spectral decomposition of the corresponding operator.
For a general system, states are typically not pure, but instead are represented as statistical mixtures of pure states, or mixed states, given by
density matrices
In quantum mechanics, a density matrix (or density operator) is a matrix used in calculating the probabilities of the outcomes of measurements performed on physical systems. It is a generalization of the state vectors or wavefunctions: while th ...
: self-adjoint operators of
trace one on a Hilbert space. Moreover, for general quantum mechanical systems, the effects of a single measurement can influence other parts of a system in a manner that is described instead by a
positive operator valued measure. Thus the structure both of the states and observables in the general theory is considerably more complicated than the idealization for pure states.
Probability theory
In
probability theory
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
, Hilbert spaces also have diverse applications. Here a fundamental Hilbert space is the space of
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
s on a given
probability space
In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models ...
, having class
(finite first and second
moments). A common operation in statistics is that of centering a random variable by subtracting its
expectation. Thus if
is a random variable, then
is its centering. In the Hilbert space view, this is the orthogonal projection of
onto the
kernel of the expectation operator, which a
continuous linear functional
In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces.
An operator between two normed spaces is a bounded linear ...
on the Hilbert space (in fact, the inner product with the constant random variable 1), and so this kernel is a closed subspace.
The
conditional expectation
In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value evaluated with respect to the conditional probability distribution. If the random variable can take on ...
has a natural interpretation in the Hilbert space. Suppose that a probability space
is given, where
is a
sigma algebra
Sigma ( ; uppercase Σ, lowercase σ, lowercase in word-final position ς; ) is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. In general mathematics, uppercase Σ is used as an operato ...
on the set
, and
is a
probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a σ-algebra that satisfies Measure (mathematics), measure properties such as ''countable additivity''. The difference between a probability measure an ...
on the measure space
. If
is a sigma subalgebra of
, then the conditional expectation