Hilbert space

# Hilbert space

Overview
The mathematical
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

concept of a Hilbert space, named after David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...

, generalizes the notion of Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

. It extends the methods of vector algebra
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...

and calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space is an abstract vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

possessing the structure
Mathematical structure
In mathematics, a structure on a set, or more generally a type, consists of additional mathematical objects that in some manner attach to the set, making it easier to visualize or work with, or endowing the collection with meaning or significance....

of an inner product
Inner product space
In mathematics, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors...

that allows length and angle to be measured. Furthermore, Hilbert spaces are required to be complete, a property that stipulates the existence of enough limits
Limit (mathematics)
In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...

in the space to allow the techniques of calculus to be used.

Hilbert spaces arise naturally and frequently in mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, and engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

, typically as infinite-dimensional function space
Function space
In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications it is a topological space, a vector space, or both.-Examples:...

s.
Discussion

Encyclopedia
The mathematical
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

concept of a Hilbert space, named after David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...

, generalizes the notion of Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

. It extends the methods of vector algebra
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...

and calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space is an abstract vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

possessing the structure
Mathematical structure
In mathematics, a structure on a set, or more generally a type, consists of additional mathematical objects that in some manner attach to the set, making it easier to visualize or work with, or endowing the collection with meaning or significance....

of an inner product
Inner product space
In mathematics, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors...

that allows length and angle to be measured. Furthermore, Hilbert spaces are required to be complete, a property that stipulates the existence of enough limits
Limit (mathematics)
In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...

in the space to allow the techniques of calculus to be used.

Hilbert spaces arise naturally and frequently in mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, and engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

, typically as infinite-dimensional function space
Function space
In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications it is a topological space, a vector space, or both.-Examples:...

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
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...

, Erhard Schmidt
Erhard Schmidt
Erhard Schmidt was a German mathematician whose work significantly influenced the direction of mathematics in the twentieth century. He was born in Tartu, Governorate of Livonia . His advisor was David Hilbert and he was awarded his doctorate from Georg-August University of Göttingen in 1905...

, and Frigyes Riesz
Frigyes Riesz
Frigyes Riesz was a mathematician who was born in Győr, Hungary and died in Budapest, Hungary. He was rector and professor at University of Szeged...

. They are indispensable tools in the theories of partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...

s, quantum mechanics
Mathematical formulation of quantum mechanics
The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. Such are distinguished from mathematical formalisms for theories developed prior to the early 1900s by the use of abstract mathematical structures, such as...

, Fourier analysis (which includes applications to signal processing
Signal processing
Signal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...

and heat transfer) and ergodic theory
Ergodic theory
Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....

which forms the mathematical underpinning of the study of thermodynamics
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...

. John von Neumann
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...

coined the term "Hilbert space" for the abstract concept underlying many of these diverse applications. The success of Hilbert space methods ushered in a very fruitful era for functional analysis
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 and the linear operators acting upon these spaces and respecting these structures in a suitable sense...

. Apart from the classical Euclidean spaces, examples of Hilbert spaces include spaces of square-integrable functions
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...

, spaces of sequences
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 numbers to the field K of real or complex numbers...

, Sobolev space
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 itself as well as its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, thus a Banach space...

s consisting of generalized function
Generalized function
In mathematics, generalized functions are objects generalizing the notion of functions. There is more than one recognized theory. Generalized functions are especially useful in making discontinuous functions more like smooth functions, and describing physical phenomena such as point charges...

s, and Hardy space
Hardy space
In complex analysis, the Hardy spaces 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...

s of holomorphic function
Holomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...

s.

Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the Pythagorean theorem
Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...

and parallelogram law
Parallelogram law
In mathematics, the simplest form of the parallelogram law 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 squares of the lengths of the two diagonals...

hold in a Hilbert space. At a deeper level, perpendicular projection onto a subspace (the analog of "dropping the altitude
Altitude (triangle)
In geometry, an altitude of a triangle is a straight line through a vertex and perpendicular to a line containing the base . This line containing the opposite side is called the extended base of the altitude. The intersection between the extended base and the altitude is called the foot of the...

" 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 (an orthonormal basis
Orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for inner product space V with finite dimension is a basis for V whose vectors are orthonormal. For example, the standard basis for a Euclidean space Rn is an orthonormal basis, where the relevant inner product is the dot product of...

), in analogy with Cartesian coordinates in the plane. When that set of axes is countably infinite, this means that the Hilbert space can also usefully be thought of in terms of infinite sequences that are square-summable. Linear operators 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
Spectral theory
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of...

.

### Motivating example: Euclidean space

One of the most familiar examples of a Hilbert space is the Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

consisting of three-dimensional vectors, denoted by R3, and equipped with the dot product
Dot product
In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...

. The dot product takes two vectors x and y, and produces a real number x·y. If x and y are represented in Cartesian coordinates, then the dot product is defined by
The dot product satisfies the properties:
1. It is symmetric in x and y: x·y = y·x.
2. It is linear
Linear function
In mathematics, the term linear function can refer to either of two different but related concepts:* a first-degree polynomial function of one variable;* a map between two vector spaces that preserves vector addition and scalar multiplication....

in its first argument: (ax1 + bx2y = ax1·y + bx2·y for any scalars a, b, and vectors x1, x2, and y.
3. It is positive definite
Definite bilinear form
In mathematics, a definite bilinear form is a bilinear form B over some vector space V such that the associated quadratic formQ=B \,...

: for all vectors x, x·x ≥ 0 with equality if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....

x = 0.

An operation on pairs of vectors that, like the dot product, satisfies these three properties is known as a (real) inner product. A vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

equipped with such an inner product is known as a (real) inner product space
Inner product space
In mathematics, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors...

. 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 (mathematics)
In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...

) of a vector, denoted ||x||, and to the angle θ between two vectors x and y by means of the formula

Multivariable calculus
Multivariable calculus
Multivariable calculus is the extension of calculus in one variable to calculus in more than one variable: the differentiated and integrated functions involve multiple variables, rather than just one....

in Euclidean space relies on the ability to compute limits
Limit (mathematics)
In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...

, and to have useful criteria for concluding that limits exist. A mathematical series
Series (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....

consisting of vectors in R3 is absolutely convergent
Absolute convergence
In mathematics, a series of numbers is said to converge absolutely if the sum of the absolute value of the summand or integrand is finite...

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 L in the Euclidean space, in the sense that
This property expresses the completeness of Euclidean space: that a series which converges absolutely also converges in the ordinary sense.

### Definition

A Hilbert space H is a real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

or complex
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

inner product space
Inner product space
In mathematics, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors...

that is also a complete metric space with respect to the distance function induced by the inner product. To say that H is a complex inner product space means that H is a complex vector space on which there is an inner product ⟨x,y⟩ associating a complex number to each pair of elements x,y of H that satisfies the following properties:
• y,x⟩ is the complex conjugate
Complex conjugate
In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...

of ⟨x,y⟩:
• x,y⟩ is linear
Linear functional
In linear algebra, a linear functional or linear form is a linear map from a vector space to its field of scalars.  In Rn, if vectors are represented as column vectors, then linear functionals are represented as row vectors, and their action on vectors is given by the dot product, or the...

in its first argument. For all complex numbers a and b,
• The inner product ⟨•,•⟩ is positive definite
Definite bilinear form
In mathematics, a definite bilinear form is a bilinear form B over some vector space V such that the associated quadratic formQ=B \,...

:
where the case of equality holds precisely when x = 0.

It follows from properties 1 and 2 that a complex inner product is antilinear in its second argument, meaning that
A real inner product space is defined in the same way, except that H is a real vector space and the inner product takes real values. Such an inner product will be bilinear: that is, linear in each argument.

The norm
Norm (mathematics)
In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...

defined by the inner product ⟨•,•⟩ is the real-valued function
and the distance between two points x,y in H is defined in terms of the norm by
That this function is a distance function means (1) that it is symmetric in x and y, (2) that the distance between x and itself is zero, and otherwise the distance between x and y must be positive, and (3) that the triangle inequality
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 ....

holds, meaning that the length of one leg of a triangle xyz 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
Cauchy–Schwarz inequality
In mathematics, the Cauchy–Schwarz inequality , is a useful inequality encountered in many different settings, such as linear algebra, analysis, probability theory, and other areas...

, which asserts
with equality if and only if x and y are linearly dependent
Linear independence
In linear algebra, a family of vectors is linearly independent if none of them can be written as a linear combination of finitely many other vectors in the collection. A family of vectors which is not linearly independent is called linearly dependent...

.

Relative to a distance function defined in this way, any inner product space is a metric space
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...

, and sometimes is known as a pre-Hilbert space. Any pre-Hilbert space that is additionally also a complete
Complete space
In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M....

space is a Hilbert space. Completeness is expressed using a form of the Cauchy criterion for sequences in H: a pre-Hilbert space H is complete if every Cauchy sequence
Cauchy sequence
In mathematics, a Cauchy sequence , named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses...

converges with respect to this norm
Limit (mathematics)
In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...

to an element in the space. Completeness can be characterized by the following equivalent condition: if a series of vectors converges absolutely
Absolute convergence
In mathematics, a series of numbers is said to converge absolutely if the sum of the absolute value of the summand or integrand is finite...

in the sense that
then the series converges in H, in the sense that the partial sums converge to an element of H.

As a complete normed space, Hilbert spaces are by definition also Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...

s. As such they are topological vector space
Topological vector space
In mathematics, a topological vector space is one of the basic structures investigated in functional analysis...

s, in which topological
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

notions like the openness
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...

and closedness
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...

of subsets are well-defined. Of special importance is the notion of a closed linear subspace
Linear subspace
The concept of a linear subspace is important in linear algebra and related fields of mathematics.A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of subspaces....

of a Hilbert space which, 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
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 numbers to the field K of real or complex numbers...

2 consists of all infinite sequences z = (z1,z2,...) of complex numbers such that the series
Series (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....

converges. The inner product on 2 is defined by
with the latter series converging as a consequence of the Cauchy–Schwarz inequality.

Completeness of the space holds provided that whenever a series of elements from 2 converges absolutely (in norm), then it converges to an element of 2. The proof is basic in mathematical analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

, 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 mathematicians and physicists. In particular, the idea of an abstract linear space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

had gained some traction towards the end of the 19th century: this is a space whose elements can be added together and multiplied by scalars (such as real or complex numbers) without necessarily identifying these elements with "geometric" vectors, such as position and momentum vectors in physical systems. Other objects studied by mathematicians at the turn of the 20th century, in particular spaces of sequences (including series
Series (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....

) 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
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...

and Erhard Schmidt
Erhard Schmidt
Erhard Schmidt was a German mathematician whose work significantly influenced the direction of mathematics in the twentieth century. He was born in Tartu, Governorate of Livonia . His advisor was David Hilbert and he was awarded his doctorate from Georg-August University of Göttingen in 1905...

's study of integral equations, that two square-integrable real-valued functions f and g on an interval [a,b] have an inner product

which has many of the familiar properties of the Euclidean dot product. In particular, the idea of an orthogonal
Orthogonality
Orthogonality occurs when two things can vary independently, they are uncorrelated, or they are perpendicular.-Mathematics:In mathematics, two vectors are orthogonal if they are perpendicular, i.e., they form a right angle...

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 K is a continuous function symmetric in x and y. The resulting eigenfunction expansion expresses the function K as a series of the form

where the functions φn 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 which fail to converge in a suitable sense to a square-integrable function: the missing ingredient, which ensures convergence, is completeness.

The second development was the Lebesgue integral, an alternative to the Riemann integral
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. The Riemann integral is unsuitable for many theoretical purposes...

introduced by Henri Lebesgue
Henri Lebesgue
Henri Léon Lebesgue was a French mathematician most famous for his theory of integration, which was a generalization of the seventeenth century concept of integration—summing the area between an axis and the curve of a function defined for that axis...

in 1904. The Lebesgue integral made it possible to integrate a much broader class of functions. In 1907, Frigyes Riesz
Frigyes Riesz
Frigyes Riesz was a mathematician who was born in Győr, Hungary and died in Budapest, Hungary. He was rector and professor at University of Szeged...

and Ernst Sigismund Fischer
Ernst Sigismund Fischer
Ernst Sigismund Fischer was a mathematician born in Vienna, Austria. He worked alongside both Mertens and Minkowski at the Universities of Vienna and Zurich, respectively...

independently proved that the space L2 of square Lebesgue-integrable functions is a complete metric space. As a consequence of the interplay between geometry and completeness, the 19th century results of Joseph Fourier
Joseph Fourier
Jean Baptiste Joseph Fourier was a French mathematician and physicist best known for initiating the investigation of Fourier series and their applications to problems of heat transfer and vibrations. The Fourier transform and Fourier's Law are also named in his honour...

, Friedrich Bessel
Friedrich Bessel
-References:* John Frederick William Herschel, A brief notice of the life, researches, and discoveries of Friedrich Wilhelm Bessel, London: Barclay, 1847 -External links:...

and Marc-Antoine Parseval
Marc-Antoine Parseval
Marc-Antoine Parseval des Chênes was a French mathematician, most famous for what is now known as Parseval's theorem, which presaged the unitarity of the Fourier transform....

on trigonometric series easily carried over to these more general spaces, resulting in a geometrical and analytical apparatus now usually known as the Riesz–Fischer theorem
Riesz–Fischer theorem
In mathematics, the Riesz–Fischer theorem in real analysis refers to a number of closely related results concerning the properties of the space L2 of square integrable functions...

.

Further basic results were proved in the early 20th century. For example, the Riesz representation theorem
Riesz representation theorem
There are several well-known theorems in functional analysis known as the Riesz representation theorem. They are named in honour of Frigyes Riesz.- The Hilbert space representation theorem :...

was independently established by Maurice Fréchet and Frigyes Riesz
Frigyes Riesz
Frigyes Riesz was a mathematician who was born in Győr, Hungary and died in Budapest, Hungary. He was rector and professor at University of Szeged...

in 1907. John von Neumann
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...

coined the term abstract Hilbert space in his work on unbounded Hermitian operators
In mathematics, on a finite-dimensional inner product space, a self-adjoint operator is an operator that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose...

. Although other mathematicians such as Hermann Weyl
Hermann Weyl
Hermann Klaus Hugo Weyl was a German mathematician and theoretical physicist. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.His...

and Norbert Wiener
Norbert Wiener
Norbert Wiener was an American mathematician.A famous child prodigy, Wiener later became an early researcher in stochastic and noise processes, contributing work relevant to electronic engineering, electronic communication, and control systems.Wiener is regarded as the originator of cybernetics, a...

had already studied particular Hilbert spaces in great detail, often from a physically motivated point of view, von Neumann gave the first complete and axiomatic treatment of them. Von Neumann later used them in his seminal work on the foundations of quantum mechanics, and in his continued work with Eugene Wigner. The name "Hilbert space" was soon adopted by others, for example by Hermann Weyl in his book on quantum mechanics and the theory of groups.

The significance of the concept of a Hilbert space was underlined with the realization that it offers one of the best mathematical formulations of quantum mechanics
Mathematical formulation of quantum mechanics
The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. Such are distinguished from mathematical formalisms for theories developed prior to the early 1900s by the use of abstract mathematical structures, such as...

. In short, the states of a quantum mechanical system are vectors in a certain Hilbert space, the observables are hermitian operators on that space, the symmetries
Symmetry
Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...

of the system are unitary operator
Unitary operator
In functional analysis, a branch of mathematics, a unitary operator is a bounded linear operator U : H → H on a Hilbert space H satisfyingU^*U=UU^*=I...

s, and measurements are orthogonal projections. The relation between quantum mechanical symmetries and unitary operators provided an impetus for the development of the unitary
Unitary representation
In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π is a unitary operator for every g ∈ G...

representation theory
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...

of groups
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

, initiated in the 1928 work of Hermann Weyl. On the other hand, in the early 1930s it became clear that certain properties of classical dynamical systems can be analyzed using Hilbert space techniques in the framework of ergodic theory
Ergodic theory
Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....

.

The algebra of observable
Observable
In physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value off...

s in quantum mechanics is naturally an algebra of operators defined on a Hilbert space, according to Werner Heisenberg
Werner Heisenberg
Werner Karl Heisenberg was a German theoretical physicist who made foundational contributions to quantum mechanics and is best known for asserting the uncertainty principle of quantum theory...

's matrix mechanics
Matrix mechanics
Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925.Matrix mechanics was the first conceptually autonomous and logically consistent formulation of quantum mechanics. It extended the Bohr Model by describing how the quantum jumps...

formulation of quantum theory. Von Neumann began investigating operator algebra
Operator algebra
In functional analysis, an operator algebra is an algebra of continuous linear operators on a topological vector space with the multiplication given by the composition of mappings...

s in the 1930s, as rings
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...

of operators on a Hilbert space. The kind of algebras studied by von Neumann and his contemporaries are now known as von Neumann algebra
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. They were originally introduced by John von Neumann, motivated by his study of single operators, group...

s. In the 1940s, Israel Gelfand
Israel Gelfand
Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gel'fand, or Izrail M. Gelfand was a Soviet mathematician who made major contributions to many branches of mathematics, including group theory, representation theory and functional analysis...

, Mark Naimark
Mark Naimark
Mark Aronovich Naimark was a Soviet mathematician.He was born in Odessa, Russian Empire into a Jewish family and died in Moscow, USSR...

and Irving Segal
Irving Segal
Irving Ezra Segal was a mathematician known for work on theoretical quantum mechanics.He was at the Massachusetts Institute of Technology...

gave a definition of a kind of operator algebras called C*-algebras that on the one hand made no reference to an underlying Hilbert space, and on the other extrapolated many of the useful features of the operator algebras that had previously been studied. The spectral theorem for self-adjoint operators in particular that underlies much of the existing Hilbert space theory was generalized to C*-algebras. These techniques are now basic in abstract harmonic analysis and representation theory.

### Lebesgue spaces

Lebesgue spaces are function space
Function space
In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications it is a topological space, a vector space, or both.-Examples:...

s associated to measure spaces
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...

(X, M, μ), where X is a set, M is a σ-algebra
Sigma-algebra
In mathematics, a σ-algebra is a technical concept for a collection of sets satisfying certain properties. The main use of σ-algebras is in the definition of measures; specifically, the collection of sets over which a measure is defined is a σ-algebra...

of subsets of X, and μ is a countably additive measure on M. Let L2(X,μ) be the space of those complex-valued measurable functions on X for which the Lebesgue integral
Lebesgue integration
In mathematics, Lebesgue integration, named after French mathematician Henri Lebesgue , refers to both the general theory of integration of a function with respect to a general measure, and to the specific case of integration of a function defined on a subset of the real line or a higher...

of the square of the absolute value
Absolute value
In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...

of the function is finite, i.e., for a function in L2(X,μ),

and where functions are identified if and only if they differ only on a set of measure zero
Null set
In mathematics, a null set is a set that is negligible in some sense. For different applications, the meaning of "negligible" varies. In measure theory, any set of measure 0 is called a null set...

.

The inner product of functions f and g in L2(X,μ) is then defined as

For f and g in L2, this integral exists because of the Cauchy–Schwarz inequality, and defines an inner product on the space. Equipped with this inner product, L2 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
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. The Riemann integral is unsuitable for many theoretical purposes...

.

The Lebesgue spaces appear in many natural settings. The spaces L2(R) and L2([0,1]) of square-integrable functions with respect to the Lebesgue measure
Lebesgue measure
In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...

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 w is any positive measurable function, the space of all measurable functions f on the interval [0,1] satisfying
is called the weighted L2 space L([0,1]), and w is called the weight function. The inner product is defined by
The weighted space L([0,1]) is identical with the Hilbert space L2([0,1],μ) where the measure μ of a Lebesgue-measurable set A is defined by
Weighted L2 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
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 itself as well as its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, thus a Banach space...

s, denoted by Hs or , are Hilbert spaces. These are a special kind of function space
Function space
In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications it is a topological space, a vector space, or both.-Examples:...

in which differentiation
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

may be performed, but which (unlike other Banach spaces such as the Hölder spaces) support the structure of an inner product. Because differentiation is permitted, Sobolev spaces are a convenient setting for the theory of partial differential equations. They also form the basis of the theory of direct methods in the calculus of variations
Direct method in calculus of variations
In the calculus of variations, a topic in mathematics, the direct method is a general method for constructing a proof of the existence of a minimizer for a given functional, introduced by Zaremba and David Hilbert around 1900. The method relies on methods of functional analysis and topology...

.

For s a non-negative integer and , the Sobolev space Hs(Ω) contains L2 functions whose weak derivative
Weak derivative
In mathematics, a weak derivative is a generalization of the concept of the derivative of a function for functions not assumed differentiable, but only integrable, i.e. to lie in the Lebesgue space L^1. See distributions for an even more general definition.- Definition :Let u be a function in the...

s of order up to s are also L2. The inner product in Hs(Ω) 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 s 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 Ω is a suitable domain, then one can define the Sobolev space Hs(Ω) as the space of Bessel potential
Bessel potential
In mathematics, the Bessel potential is a potential similar to the Riesz potential but with better decay properties at infinity....

s; roughly,
Here Δ is the Laplacian and (1 − Δ)s/2 is understood in terms of the spectral mapping theorem. Apart from providing a workable definition of Sobolev spaces for non-integer s, this definition also has particularly desirable properties under the Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

that make it ideal for the study of pseudodifferential operators. Using these methods on a compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

Riemannian manifold
Riemannian manifold
In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...

, one can obtain for instance the Hodge decomposition which is the basis of Hodge theory
Hodge theory
In mathematics, Hodge theory, named after W. V. D. Hodge, is one aspect of the study of the algebraic topology of a smooth manifold M. More specifically, it works out the consequences for the cohomology groups of M, with real coefficients, of the partial differential equation theory of generalised...

.

### Spaces of holomorphic functions

Hardy spaces
The Hardy space
Hardy space
In complex analysis, the Hardy spaces 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...

s are function spaces, arising in complex analysis
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 useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...

and harmonic analysis
Harmonic analysis
Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms...

, whose elements are certain holomorphic function
Holomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...

s in a complex domain. Let U denote the unit disc in the complex plane. Then the Hardy space H2(U) is defined to be the space of holomorphic functions f on U 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 f is in H2(U) if and only if

where

Thus H2(U) consists of those functions which are L2 on the circle, and whose negative frequency Fourier coefficients vanish.

Bergman spaces
The Bergman space
Bergman space
In complex analysis, a branch of mathematics, a Bergman space, named after Stefan Bergman, is a function space of holomorphic functions in a domain D of the complex plane that are sufficiently well-behaved at the boundary that they are absolutely integrable...

s are another family of Hilbert spaces of holomorphic functions. Let D be a bounded open set in the complex plane
Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...

(or a higher dimensional complex space) and let L2,h(D) be the space of holomorphic functions ƒ in D that are also in L2(D) in the sense that
where the integral is taken with respect to the Lebesgue measure in D. Clearly L2,h(D) is a subspace of L2(D); in fact, it is a closed
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...

subspace, and so a Hilbert space in its own right. This is a consequence of the estimate, valid on compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

subsets K of D, that
which in turn follows from Cauchy's integral formula
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 of the disk, and it provides integral formulas for all...

. Thus convergence of a sequence of holomorphic functions in L2(D) implies also compact convergence
Compact convergence
In mathematics compact convergence is a type of convergence which generalizes the idea of uniform convergence. It is associated with the compact-open topology.-Definition:...

, and so the limit function is also holomorphic. Another consequence of this inequality is that the linear functional that evaluates a function ƒ at a point of D is actually continuous on L2,h(D). The Riesz representation theorem implies that the evaluation functional can be represented as an element of L2,h(D). Thus, for every z ∈ D, there is a function ηz ∈ L2,h(D) such that
for all ƒ ∈ L2,h(D). The integrand
is known as the Bergman kernel
Bergman kernel
In the mathematical study of several complex variables, the Bergman kernel, named after Stefan Bergman, is a reproducing kernel for the Hilbert space of all square integrable holomorphic functions on a domain D in Cn....

of D. This integral kernel satisfies a reproducing property

A Bergman space is an example of a reproducing kernel Hilbert space
Reproducing kernel Hilbert space
In functional analysis , a reproducing kernel Hilbert space is a Hilbert space of functions in which pointwise evaluation is a continuous linear functional. Equivalently, they are spaces that can be defined by reproducing kernels...

, which is a Hilbert space of functions along with a kernel K(ζ,z) that verifies a reproducing property analogous to this one. The Hardy space H2(D) also admits a reproducing kernel, known as the Szegő kernel
Szegő kernel
In the mathematical study of several complex variables, the Szegő kernel is an integral kernel that gives rise to a reproducing kernel on a natural Hilbert space of holomorphic functions...

. Reproducing kernels are common in other areas of mathematics as well. For instance, in harmonic analysis
Harmonic analysis
Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms...

the Poisson kernel
Poisson kernel
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 disc. The kernel can be understood as the derivative of the Green's function for the Laplace equation...

is a reproducing kernel for the Hilbert space of square-integrable harmonic function
Harmonic function
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R which satisfies Laplace's equation, i.e....

s in the unit ball. That the latter is a Hilbert space at all is a consequence of the mean value theorem for harmonic functions.

## Applications

Many of the applications of Hilbert spaces exploit the fact that Hilbert spaces support generalizations of simple geometric concepts like projection and change of basis
Change of basis
In linear algebra, change of basis refers to the conversion of vectors and linear transformations between matrix representations which have different bases.-Expression of a basis:...

from their usual finite dimensional setting. In particular, the spectral theory
Spectral theory
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of...

of continuous
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

In mathematics, on a finite-dimensional inner product space, a self-adjoint operator is an operator that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose...

linear operators on a Hilbert space generalizes the usual spectral decomposition of a matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

, 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
Ordinary differential equation
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....

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. The problem is a differential equation of the form
for an unknown function y on an interval [a,b], satisfying general homogeneous Robin boundary conditions
The functions p, q, and w are given in advance, and the problem is to find the function y and constants λ for which the equation has a solution. The problem only has solutions for certain values of λ, called eigenvalues of the system, and this is a consequence of the spectral theorem for compact operator
Compact operator
In functional analysis, a branch of mathematics, a compact operator is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset of Y...

s applied to the integral operator defined by the Green's function
Green's function
In mathematics, a Green's function is a type of function used to solve inhomogeneous differential equations subject to specific initial conditions or boundary conditions...

for the system. Furthermore, another consequence of this general result is that the eigenvalues λ of the system can be arranged in an increasing sequence tending to infinity.

### Partial differential equations

Hilbert spaces form a basic tool in the study of partial differential equations. For many classes of partial differential equations, such as linear elliptic equations, it is possible to consider a generalized solution (known as a weak
Weak derivative
In mathematics, a weak derivative is a generalization of the concept of the derivative of a function for functions not assumed differentiable, but only integrable, i.e. to lie in the Lebesgue space L^1. See distributions for an even more general definition.- Definition :Let u be a function in the...

solution) by enlarging the class of functions. Many weak formulations involve the class of Sobolev functions
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 itself as well as its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, thus a Banach space...

, 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
Galerkin method
In mathematics, in the area of numerical analysis, Galerkin methods are a class of methods for converting a continuous operator problem to a discrete problem. In principle, it is the equivalent of applying the method of variation of parameters to a function space, by converting the equation to a...

(a finite element method
Finite element method
The finite element method is a numerical technique for finding approximate solutions of partial differential equations as well as integral equations...

) for numerical solution of partial differential equations.

A typical example is the Poisson equation  with Dirichlet boundary conditions in a bounded domain Ω in R2. The weak formulation consists of finding a function u such that, for all continuously differentiable functions v in Ω vanishing on the boundary:

This can be recast in terms of the Hilbert space H(Ω) consisting of functions u such that u, along with its weak partial derivatives, are square integrable on Ω, and which vanish on the boundary. The question then reduces to finding u in this space such that for all v in this space

where a is a continuous bilinear form, and b is a continuous linear functional
Linear functional
In linear algebra, a linear functional or linear form is a linear map from a vector space to its field of scalars.  In Rn, if vectors are represented as column vectors, then linear functionals are represented as row vectors, and their action on vectors is given by the dot product, or the...

, given respectively by

Since the Poisson equation is elliptic, it follows from Poincaré's inequality that the bilinear form a is coercive
Coercive function
In mathematics, a coercive function is a function that "grows rapidly" at the extremes of the space on which it is defined. More precisely, a function f : Rn → Rn is called coercive if...

. 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
Parabolic partial differential equation
A parabolic partial differential equation is a type of second-order partial differential equation , describing a wide family of problems in science including heat diffusion, ocean acoustic propagation, in physical or mathematical systems with a time variable, and which behave essentially like heat...

s and certain hyperbolic partial differential equation
Hyperbolic partial differential equation
In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation that, roughly speaking, has a well-posed initial value problem for the first n−1 derivatives. More precisely, the Cauchy problem can be locally solved for arbitrary initial data along...

s.

### Ergodic theory

The field of ergodic theory
Ergodic theory
Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....

is the study of the long-term behavior of chaotic
Chaos theory
Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...

dynamical system
Dynamical system
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...

s. The protypical case of a field to which ergodic theory is applicable is that of thermodynamics
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...

in which, although 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
Laws of thermodynamics
The four laws of thermodynamics summarize its most important facts. They define fundamental physical quantities, such as temperature, energy, and entropy, in order to describe thermodynamic systems. They also describe the transfer of energy as heat and work in thermodynamic processes...

are assertions about such average behavior. In particular, one formulation of the zeroth law of thermodynamics
Zeroth law of thermodynamics
The zeroth law of thermodynamics is a generalization principle of thermal equilibrium among bodies, or thermodynamic systems, in contact.The zeroth law states that if two systems are in thermal equilibrium with a third system, they are also in thermal equilibrium with each other.Systems are said to...

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
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

.

An ergodic dynamical system is one for which, apart from the energy—measured by the Hamiltonian
Hamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...

—there are no other functionally independent conserved quantities on the phase space
Phase space
In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...

. More explicitly, suppose that the energy E is fixed, and let ΩE be the subset of the phase space consisting of all states of energy E (an energy surface), and let Tt denote the evolution operator on the phase space. The dynamical system is ergodic if there are no continuous non-constant functions on ΩE such that
for all w on ΩE and all time t. Liouville's theorem
Liouville's theorem (Hamiltonian)
In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics...

implies that there exists a measure μ on the energy surface that is invariant under the time translation. As a result, time translation is a unitary transformation
Unitary transformation
In mathematics, a unitary transformation may be informally defined as a transformation that respects the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation....

of the Hilbert space L2E,μ) consisting of square-integrable functions on the energy surface ΩE with respect to the inner product

The von Neumann mean ergodic theorem states the following:
• If Ut is a (strongly continuous) one-parameter semigroup of unitary operators on a Hilbert space H, and P is the orthogonal projection onto the space of common fixed points of Ut, {xH | Utx = x for all t > 0}, 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 ƒ ∈ L2E,μ),
That is, the long time average of an observable ƒ is equal to its expectation value over an energy surface.

### Fourier analysis

One of the basic goals of Fourier analysis is to decompose a function into a (possibly infinite) linear combination
Linear combination
In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...

of given basis functions: the associated Fourier series
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...

. The classical Fourier series associated to a function ƒ defined on the interval [0,1] 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 λ/n (n=integer) shorter than the wavelength λ of the sawtooth itself (except for n=1, 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
Gibbs phenomenon
In mathematics, the Gibbs phenomenon, named after the American physicist J. Willard Gibbs, is the peculiar manner in which the Fourier series of a piecewise continuously differentiable periodic function behaves at a jump discontinuity: the nth partial sum of the Fourier series has large...

.

A significant problem in classical Fourier series asks in what sense the Fourier series converges, if at all, to the function ƒ. Hilbert space methods provide one possible answer to this question. The functions en(θ) = e2πinθ form an orthogonal basis of the Hilbert space L2([0,1]). 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 L2 mean).

The problem can also be studied from the abstract point of view: every Hilbert space has an orthonormal basis
Orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for inner product space V with finite dimension is a basis for V whose vectors are orthonormal. For example, the standard basis for a Euclidean space Rn is an orthonormal basis, where the relevant inner product is the dot product of...

, 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 L2([0,1]). In many circumstances, it is desirable not to decompose a function into trigonometric functions, but rather into orthogonal polynomials
Orthogonal polynomials
In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials, and consist of the Hermite polynomials, the Laguerre polynomials, the Jacobi polynomials together with their special cases the ultraspherical polynomials, the Chebyshev polynomials, and the...

or wavelet
Wavelet
A wavelet is a wave-like oscillation with an amplitude that starts out at zero, increases, and then decreases back to zero. It can typically be visualized as a "brief oscillation" like one might see recorded by a seismograph or heart monitor. Generally, wavelets are purposefully crafted to have...

s for instance, and in higher dimensions into spherical harmonics
Spherical harmonics
In mathematics, spherical harmonics are the angular portion of a set of solutions to Laplace's equation. Represented in a system of spherical coordinates, Laplace's spherical harmonics Y_\ell^m are a specific set of spherical harmonics that forms an orthogonal system, first introduced by Pierre...

.

For instance, if en are any orthonormal basis functions of L2[0,1], then a given function in L2[0,1] can be approximated as a finite linear combination
The coefficients {aj} are selected to make the magnitude of the difference ||||2 as small as possible. Geometrically, the best approximation is the orthogonal projection of ƒ onto the subspace consisting of all linear combinations of the {ej}, and can be calculated by
That this formula minimizes the difference ||||2 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
Eigenfunction
In mathematics, an eigenfunction of a linear operator, A, defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. More precisely, one has...

s of a differential operator
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, accepting a function and returning another .This article considers only linear operators,...

(typically the Laplace operator
Laplace operator
In mathematics the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols ∇·∇, ∇2 or Δ...

): this forms the foundation for the spectral study of functions, in reference to the spectrum
Spectral theorem
In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides conditions under which an operator or a matrix can be diagonalized...

of the differential operator. A concrete physical application involves the problem of hearing the shape of a drum
Hearing the shape of a drum
To hear the shape of a drum is to infer information about the shape of the drumhead from the sound it makes, i.e., from the list of basic harmonics, via the use of mathematical theory...

: given the fundamental modes of vibration that a drumhead is capable of producing, can one infer the shape of the drum itself? The mathematical formulation of this question involves the Dirichlet eigenvalue
Dirichlet eigenvalue
In mathematics, the Dirichlet eigenvalues are the fundamental modes of vibration of an idealized drum with a given shape. The problem of whether one can hear the shape of a drum is: given the Dirichlet eigenvalues, what features of the shape of the drum can one deduce. Here a "drum" is thought of...

s of the Laplace equation in the plane, that represent the fundamental modes of vibration in direct analogy with the integers that represent the fundamental modes of vibration of the violin string.

Spectral theory also underlies certain aspects of the Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

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
Continuous spectrum
The spectrum of a linear operator is commonly divided into three parts: point spectrum, continuous spectrum, and residual spectrum.If H is a topological vector space and A:H \to H is a linear map, the spectrum of A is the set of complex numbers \lambda such that A - \lambda I : H \to H is not...

of the Laplacian. The Fourier transformation is also geometrical, in a sense made precise by the Plancherel theorem
Plancherel theorem
In mathematics, the Plancherel theorem is a result in harmonic analysis, proved by Michel Plancherel in 1910. It states that the integral of a function's squared modulus is equal to the integral of the squared modulus of its frequency spectrum....

, that asserts that it is an isometry
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...

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
Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms...

, as evidenced for instance by the Plancherel theorem for spherical functions
Plancherel theorem for spherical functions
In mathematics, the Plancherel theorem for spherical functions is an important result in the representation theory of semisimple Lie groups, due in its final form to Harish-Chandra...

occurring in noncommutative harmonic analysis
Noncommutative harmonic analysis
In mathematics, noncommutative harmonic analysis is the field in which results from Fourier analysis are extended to topological groups which are not commutative...

.

### Quantum mechanics

In the mathematically rigorous formulation of quantum mechanics, developed by Paul Dirac
Paul Dirac
Paul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...

and John von Neumann
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...

, 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
State space (physics)
In physics, a state space is a complex Hilbert space within which the possible instantaneous states of the system may be described by a unit vector. These state vectors, using Dirac's bra-ket notation, can often be treated as vectors and operated on using the rules of linear algebra...

, well defined up to a complex number of norm 1 (the phase factor). In other words, the possible states are points in the projectivization
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....

of a Hilbert space, usually called the complex projective space
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 complex projective space label the complex lines...

. 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
Spinors in three dimensions
In mathematics, the spinor concept as specialised to three dimensions can be treated by means of the traditional notions of dot product and cross product. This is part of the detailed algebraic discussion of the rotation group SO.-Formulation:...

. Each observable is represented by a self-adjoint
In mathematics, on a finite-dimensional inner product space, a self-adjoint operator is an operator that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose...

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 time evolution of a quantum state is described by the Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....

, in which the Hamiltonian
Hamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...

, the operator
Operator (physics)
In physics, an operator is a function acting on the space of physical states. As a resultof its application on a physical state, another physical state is obtained, very often along withsome extra relevant information....

corresponding to the total energy of the system, generates time evolution.

The inner product between two state vectors is a complex number known as a probability amplitude
Probability amplitude
In quantum mechanics, a probability amplitude is a complex number whose modulus squared represents a probability or probability density.For example, if the probability amplitude of a quantum state is \alpha, the probability of measuring that state is |\alpha|^2...

. 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
Absolute value
In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...

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
Density matrix
In quantum mechanics, a density matrix is a self-adjoint positive-semidefinite matrix of trace one, that describes the statistical state of a quantum system...

: 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.

Heisenberg's uncertainty principle
Uncertainty principle
In quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...

is represented by the statement that the operators corresponding to certain observables do not commute, and gives a specific form that the commutator
Commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.-Group theory:...

must have.

### Pythagorean identity

Two vectors u and v in a Hilbert space H are orthogonal when  = 0. The notation for this is . More generally, when S is a subset in H, the notation means that u is orthogonal to every element from S.

When u and v are orthogonal, one has

By induction on n, this is extended to any family u1,...,un of n orthogonal vectors,

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 Σ uk of orthogonal vectors converges in H  if and only if the series of squares of norms converges, and
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
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...

. Furthermore, in every Hilbert space the following parallelogram identity holds:

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
Polarization identity
In mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space. Let \|x\| \, denote the norm of vector x and \langle x, \ y \rangle \, the inner product of vectors x and y...

. For real Hilbert spaces, the polarization identity is
For complex Hilbert spaces, it is
The parallelogram law implies that any Hilbert space is a uniformly convex Banach space.

### Best approximation

If C is a non-empty closed convex subset of a Hilbert space H and x a point in H, there exists a unique point y ∈ C which minimizes the distance between x and points in C,

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

When this result is applied to a closed subspace F of H, it can be shown that the point y ∈ F closest to x is characterized by

This point y is the orthogonal projection of x onto F, and the mapping PF : is linear (see Orthogonal complements and projections). This result is especially significant in applied mathematics
Applied mathematics
Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...

, especially numerical analysis
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....

, where it forms the basis of least squares
Least squares
The method of least squares is a standard approach to the approximate solution of overdetermined systems, i.e., sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in solving every...

methods.

In particular, when F is not equal to H, one can find a non-zero vector v orthogonal to F
(select x not in F and v = . A very useful criterion is obtained by applying this observation to the closed subspace F generated by a subset S of H.
A subset S of H spans a dense vector subspace if (and only if) the vector 0 is the sole vector v ∈ H orthogonal to S.

### Duality

The dual space H is the space of all continuous linear functions from the space H into the base field. It carries a natural norm, defined by
This norm satisfies the parallelogram law, and so the dual space is also an inner product space. The dual space is also complete, and so it is a Hilbert space in its own right.

The Riesz representation theorem
Riesz representation theorem
There are several well-known theorems in functional analysis known as the Riesz representation theorem. They are named in honour of Frigyes Riesz.- The Hilbert space representation theorem :...

affords a convenient description of the dual. To every element u of H, there is a unique element φu of H, defined by
The mapping is an antilinear mapping from H to H. The Riesz representation theorem states that this mapping is an antilinear isomorphism. Thus to every element φ of the dual H there exists one and only one uφ in H such that
for all x ∈ H. The inner product on the dual space H satisfies
The reversal of order on the right-hand side restores linearity in φ from the antilinearity of uφ. In the real case, the antilinear isomorphism from H to its dual is actually an isomorphism, and so real Hilbert spaces are naturally isomorphic to their own duals.

The representing vector uφ is obtained in the following way. When φ ≠ 0, the kernel
Kernel (algebra)
In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. An important special case is the kernel of a matrix, also called the null space.The definition of kernel takes...

F = ker φ is a closed vector subspace of H, not equal to H, hence there exists a non-zero vector v orthogonal to F. The vector u is a suitable scalar multiple λv of v. The requirement that φ(v) = ⟨vu⟩ yields

This correspondence φu is exploited by the bra-ket notation
Bra-ket notation
Bra-ket notation is a standard notation for describing quantum states in the theory of quantum mechanics composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics...

popular in physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

. It is common in physics to assume that the inner product, denoted by ⟨x|y⟩, is linear on the right,
The result ⟨x|y⟩ can be seen as the action of the linear functional ⟨x| (the bra) on the vector  |y⟩ (the ket).

The Riesz representation theorem relies fundamentally not just on the presence of an inner product, but also on the completeness of the space. In fact, the theorem implies that the topological dual
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...

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 H is reflexive
Reflexive space
In functional analysis, a Banach space is called reflexive if it coincides with the dual of its dual space in the topological and algebraic senses. Reflexive Banach spaces are often characterized by their geometric properties.- Normed spaces :Suppose X is a normed vector space over R or C...

, meaning that the natural map from H into its double dual space
Dual space
In mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra...

is an isomorphism.

### Weakly convergent sequences

In a Hilbert space H, a sequence {xn} is weakly convergent to a vector x ∈ H when

for every .

For example, any orthonormal sequence {ƒn} converges weakly to 0, as a consequence of Bessel's inequality. Every weakly convergent sequence {xn} is bounded, by the uniform boundedness principle
Uniform boundedness principle
In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerstones of the field...

.

Conversely, every bounded sequence in a Hilbert space admits weakly convergent subsequences (Alaoglu's theorem). This fact may be used to prove minimization results for continuous convex function
Convex function
In mathematics, a real-valued function f defined on an interval is called convex if the graph of the function lies below the line segment joining any two points of the graph. Equivalently, a function is convex if its epigraph is a convex set...

als, in the same way that the Bolzano–Weierstrass theorem
Bolzano–Weierstrass theorem
In real analysis, the Bolzano–Weierstrass theorem is a fundamental result about convergence in a finite-dimensional Euclidean space Rn. The theorem states thateach bounded sequence in Rn has a convergent subsequence...

is used for continuous functions on Rd. Among several variants, one simple statement is as follows:
If ƒ : is a convex continuous function such that ƒ(x) tends to +∞ when ||x|| tends to ∞, then ƒ admits a minimum at some point .

This fact (and its various generalizations) are fundamental for direct methods in the calculus of variations
Calculus of variations
Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...

. 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 H are weakly compact
Weak topology
In mathematics, weak topology is an alternative term for initial topology. The term is most commonly used for the initial topology of a topological vector space with respect to its continuous dual...

, since H is reflexive. The existence of weakly convergent subsequences is a special case of the Eberlein–Šmulian theorem
Eberlein–Šmulian theorem
In the mathematical field of functional analysis, the Eberlein–Šmulian theorem is a result relating three different kinds of weak compactness in a Banach space...

.

### Banach space properties

Any general property of Banach spaces
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...

continues to hold for Hilbert spaces. The open mapping theorem
Open mapping theorem (functional analysis)
In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem , is a fundamental result which states that if a continuous linear operator between Banach spaces is surjective then it is an open map...

states that a continuous
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

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
Bounded inverse theorem
In mathematics, the bounded inverse theorem is a result in the theory of bounded linear operators on Banach spaces. It states that a bijective bounded linear operator T from one Banach space to another has bounded inverse T−1. It is equivalent to both the open mapping theorem and the closed...

, 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
Closed graph theorem
In mathematics, the closed graph theorem is a basic result in functional analysis which characterizes continuous linear operators between Banach spaces in terms of the operator graph.- The closed graph theorem :...

, which asserts that a function from one Banach space to another is continuous if and only if its graph is a closed set
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...

. In the case of Hilbert spaces, this is basic in the study of unbounded operator
Unbounded operator
In mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases....

s (see closed operator
Closed operator
In mathematics, specifically in functional analysis, closed linear operators are an important class of linear operators on Banach spaces. They are more general than bounded operators, and therefore not necessarily continuous, but they still retain nice enough properties that one can define the...

).

The (geometrical) Hahn–Banach theorem
Hahn–Banach theorem
In mathematics, the Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed...

asserts that a closed convex set can be separated from any point outside it by means of a hyperplane
Hyperplane
A hyperplane is a concept in geometry. It is a generalization of the plane into a different number of dimensions.A hyperplane of an n-dimensional space is a flat subset with dimension n − 1...

of the Hilbert space. This is an immediate consequence of the best approximation property: if y is the element of a closed convex set F closest to x, then the separating hyperplane is the plane perpendicular to the segment xy passing through its midpoint.

### Bounded operators

The continuous linear operators A : H1H2 from a Hilbert space H1 to a second Hilbert space H2 are bounded in the sense that they map bounded set
Bounded set
In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. Conversely, a set which is not bounded is called unbounded...

s to bounded sets. Conversely, if an operator is bounded, then it is continuous. The space of such bounded linear operators has a norm
Norm (mathematics)
In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...

, the operator norm
Operator norm
In mathematics, the operator norm is a means to measure the "size" of certain linear operators. Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces.- Introduction and definition :...

given by

The sum and the composite of two bounded linear operators is again bounded and linear. For y in H2, the map that sends x ∈ H1 to ⟨Ax, y⟩ is linear and continuous, and according to the Riesz representation theorem can therefore be represented in the form

for some vector Ay in H1.
This defines another bounded linear operator A : H2 → H1, the adjoint
In mathematics, specifically in functional analysis, each linear operator on a Hilbert space has a corresponding adjoint operator.Adjoints of operators generalize conjugate transposes of square matrices to infinite-dimensional situations...

of A. One can see that .

The set B(H) of all bounded linear operators on H, together with the addition and composition operations, the norm and the adjoint operation, is a C*-algebra, which is a type of operator algebra
Operator algebra
In functional analysis, an operator algebra is an algebra of continuous linear operators on a topological vector space with the multiplication given by the composition of mappings...

.

An element A  of B(H) is called self-adjoint or Hermitian if A = A. If A  is Hermitian and 0 for every x, then A  is called non-negative, written A ≥ 0; if equality holds only when x = 0, then A  is called positive. The set of self adjoint operators admits a partial order, in which A ≥ B if A − B ≥ 0. If A  has the form BB  for some B, then A  is non-negative; if B is invertible, then A  is positive. A converse is also true in the sense that, for a non-negative operator A, there exists a unique non-negative square root
Square root of a matrix
In mathematics, the square root of a matrix extends the notion of square root from numbers to matrices. A matrix B is said to be a square root of A if the matrix product B · B is equal to A.-Properties:...

B such that

In a sense made precise by the spectral theorem, self-adjoint operators can usefully be thought of as operators that are "real". An element A  of B(H) is called normal if AA = A A. Normal operators decompose into the sum of a self-adjoint operators and an imaginary multiple of a self adjoint operator
that commute with each other. Normal operators can also usefully be thought of in terms of their real and imaginary parts.

An element U  of B(H) is called unitary
Unitary operator
In functional analysis, a branch of mathematics, a unitary operator is a bounded linear operator U : H → H on a Hilbert space H satisfyingU^*U=UU^*=I...

if U  is invertible and its inverse is given by U. This can also be expressed by requiring that U  be onto for all x and y in H. The unitary operators form a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

under composition, which is the isometry group
Isometry group
In mathematics, the isometry group of a metric space is the set of all isometries from the metric space onto itself, with the function composition as group operation...

of H.

An element of B(H) is compact
Compact operator
In functional analysis, a branch of mathematics, a compact operator is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset of Y...

if it sends bounded sets to relatively compact sets. Equivalently, a bounded operator T is compact if, for any bounded sequence {xk}, the sequence {Txk} 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 equation
Integral equation
In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. There is a close connection between differential and integral equations, and some problems may be formulated either way...

s. Fredholm operator
Fredholm operator
In mathematics, a Fredholm operator is an operator that arises in the Fredholm theory of integral equations. It is named in honour of Erik Ivar Fredholm....

s are those which differ from a compact operator by a multiple of the identity, and are equivalently characterized as operators with a finite dimensional kernel and cokernel
Cokernel
In mathematics, the cokernel of a linear mapping of vector spaces f : X → Y is the quotient space Y/im of the codomain of f by the image of f....

. The index of a Fredholm operator T is defined by
The index is homotopy
Homotopy
In topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions...

invariant, and plays a deep role in differential geometry via the Atiyah–Singer index theorem
Atiyah–Singer index theorem
In differential geometry, the Atiyah–Singer index theorem, proved by , states that for an elliptic differential operator on a compact manifold, the analytical index is equal to the topological index...

.

### Unbounded operators

Unbounded operator
Unbounded operator
In mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases....

s are also tractable in Hilbert spaces, and have important applications to quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

. An unbounded operator T on a Hilbert space H is defined to be a linear operator whose domain D(T) is a linear subspace of H. Often the domain D(T) is a dense subspace of H, in which case T is known as a densely defined operator.

The adjoint of a densely defined unbounded operator is defined in essentially the same manner as for bounded operators. Self-adjoint unbounded operators
In mathematics, on a finite-dimensional inner product space, a self-adjoint operator is an operator that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose...

play the role of the observables in the mathematical formulation of quantum mechanics. Examples of self-adjoint unbounded operators on the Hilbert space L2(R) are:
• A suitable extension of the differential operator

where i is the imaginary unit and f is a differentiable function of compact support.
• The multiplication-by-x operator:

These correspond to the momentum
Momentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

and position
Position operator
In quantum mechanics, the position operator is the operator that corresponds to the position observable of a particle. Consider, for example, the case of a spinless particle moving on a line. The state space for such a particle is L2, the Hilbert space of complex-valued and square-integrable ...

observables, respectively. Note that neither A nor B is defined on all of H, since in the case of A the derivative need not exist, and in the case of B the product function need not be square integrable. In both cases, the set of possible arguments form dense subspaces of L2(R).

### Direct sums

Two Hilbert spaces H1 and H2 can be combined into another Hilbert space, called the (orthogonal) direct sum, and denoted
consisting of the set of all ordered pair
Ordered pair
In mathematics, an ordered pair is a pair of mathematical objects. In the ordered pair , the object a is called the first entry, and the object b the second entry of the pair...

s (x1x2) where , , and inner product defined by

More generally, if Hi is a family of Hilbert spaces indexed by , then the direct sum of the Hi, denoted
consists of the set of all indexed families
in the Cartesian product
Cartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...

of the Hi such that
The inner product is defined by

Each of the Hi is included as a closed subspace in the direct sum of all of the Hi. Moreover, the Hi are pairwise orthogonal. Conversely, if there is a system of closed subspaces Vi, , in a Hilbert space H which are pairwise orthogonal and whose union is dense in H, then H is canonically isomorphic to the direct sum of Vi. In this case, H is called the internal direct sum of the Vi. A direct sum (internal or external) is also equipped with a family of orthogonal projections Ei onto the ith direct summand Hi. These projections are bounded, self-adjoint, idempotent operators which satisfy the orthogonality condition

The spectral theorem
Spectral theorem
In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides conditions under which an operator or a matrix can be diagonalized...

for compact
Compact operator
In functional analysis, a branch of mathematics, a compact operator is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset of Y...

self-adjoint operators on a Hilbert space H states that H 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
Fock space
The Fock space is an algebraic system used in quantum mechanics to describe quantum states with a variable or unknown number of particles. It is named after V. A...

of a system containing a variable number of particles, where each Hilbert space in the direct sum corresponds to an additional degree of freedom for the quantum mechanical system. In representation theory
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...

, the Peter–Weyl theorem
Peter–Weyl theorem
In mathematics, the Peter–Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian. It was initially proved by Hermann Weyl, with his student Fritz Peter, in the setting of a compact topological group G...

guarantees that any unitary representation
Unitary representation
In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π is a unitary operator for every g ∈ G...

of a compact group
Compact group
In mathematics, a compact group is a topological group whose topology is compact. Compact groups are a natural generalisation of finite groups with the discrete topology and have properties that carry over in significant fashion...

on a Hilbert space splits as the direct sum of finite-dimensional representations.

### Tensor products

If H1 and H2, then one defines an inner product on the (ordinary) tensor product
Tensor product
In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...

as follows. On simple tensors, let
This formula then extends by sesquilinearity
Sesquilinear form
In mathematics, a sesquilinear form on a complex vector space V is a map V × V → C that is linear in one argument and antilinear in the other. The name originates from the numerical prefix sesqui- meaning "one and a half"...

to an inner product on . The Hilbertian tensor product of H1 and H2, 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 L2([0, 1]). The Hilbertian tensor product of two copies of L2([0, 1]) is isometrically and linearly isomorphic to the space L2([0, 1]2) of square-integrable functions on the square [0, 1]2. This isomorphism sends a simple tensor to the function
on the square.

This example is typical in the following sense. Associated to every simple tensor product is the rank one operator
from the (continuous) dual H1 to H2. This mapping defined on simple tensors extends to a linear identification between and the space of finite rank operators from H1 to H2.
This extends to a linear isometry of the Hilbertian tensor product with the Hilbert space HS(H1, H2) of Hilbert–Schmidt operators from H1 to H2.

## Orthonormal bases

The notion of an orthonormal basis
Orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for inner product space V with finite dimension is a basis for V whose vectors are orthonormal. For example, the standard basis for a Euclidean space Rn is an orthonormal basis, where the relevant inner product is the dot product of...

from linear algebra generalizes over to the case of Hilbert spaces. In a Hilbert space H, an orthonormal basis is a family {ek} of elements of H satisfying the conditions:
1. Orthogonality: Every two different elements of B are orthogonal: for all k, j in B with .
2. Normalization: Every element of the family has norm 1:||ek|| = 1 for all k in B.
3. Completeness: The linear span
Linear span
In the mathematical subfield of linear algebra, the linear span of a set of vectors in a vector space is the intersection of all subspaces containing that set...

of the family ek, , is dense
Dense set
In topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...

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 B is countable
Countable set
In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor...

). 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 S is any orthonormal set and v is orthogonal to S, then v is orthogonal to the closure of the linear span of S, which is the whole space.

Examples of orthonormal bases include:
• the set {(1,0,0), (0,1,0), (0,0,1)} forms an orthonormal basis of R3 with the dot product;
• the sequence {ƒn : nZ} with ƒn(x) = exp
Exponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...

(2πinx) forms an orthonormal basis of the complex space L2([0,1]);

In the infinite-dimensional case, an orthonormal basis will not be a basis in the sense of linear algebra
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...

; 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  2 of square-summable sequences of complex numbers is the set of infinite sequences

of complex numbers such that

This space has an orthonormal basis:

More generally, if B is any set, then one can form a Hilbert space of sequences with index set B, defined by

The summation over B is here defined by

the supremum
Supremum
In mathematics, given a subset S of a totally or partially ordered set T, the supremum of S, if it exists, is the least element of T that is greater than or equal to every element of S. Consequently, the supremum is also referred to as the least upper bound . If the supremum exists, it is unique...

being taken over all finite subsets of B. It follows that, in order for this sum to be finite, every element of  2(B) has only countably many nonzero terms. This space becomes a Hilbert space with the inner product
for all x and y in  2(B). Here the sum also has only countably many nonzero terms, and is unconditionally convergent by the Cauchy–Schwarz inequality.

An orthonormal basis of  2(B) is indexed by the set B, given by

### Bessel's inequality and Parseval's formula

Let be a finite orthonormal system in H. For an arbitrary vector x in H, let

Then  = for every k = . It follows that is orthogonal to each ƒk, hence is orthogonal to y. Using the Pythagorean identity twice, it follows that

Let , be an arbitrary orthonormal system in H. Applying the preceding inequality to every finite subset J of I gives the Bessel inequality

(according to the definition of the sum of an arbitrary family of non-negative real numbers).

Geometrically, Bessel's inequality implies that the orthogonal projection of x onto the linear subspace spanned by the fi has norm that does not exceed that of x. 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 more powerful Parseval identity which governs the case when Bessel's inequality is actually an equality. If {ek}kB is an orthonormal basis of H, then every element x of H may be written as

Even if B 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 x, and the individual coefficients ⟨x,ek⟩ are the Fourier coefficients of x. Parseval's formula is then

Conversely, if {ek} is an orthonormal set such that Parseval's identity holds for every x, then {ek} is an orthonormal basis.

### Hilbert dimension

As a consequence of Zorn's lemma
Zorn's lemma
Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory that states:Suppose a partially ordered set P has the property that every chain has an upper bound in P...

, every Hilbert space admits an orthonormal basis; furthermore, any two orthonormal bases of the same space have the same cardinality
Cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...

, called the Hilbert dimension of the space. For instance, since 2(B) has an orthonormal basis indexed by B, its Hilbert dimension is the cardinality of B (which may be a finite integer, or a countable or uncountable cardinal number
Cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...

).

As a consequence of Parseval's identity, if {ek}kB is an orthonormal basis of H, then the map  ℓ2(B) defined by is an isometric isomorphism of Hilbert spaces: it is a bijective linear mapping such that
for all x and y in H. The cardinal number
Cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...

of B is the Hilbert dimension of H. Thus every Hilbert space is isometrically isomorphic to a sequence space  ℓ2(B) for some set B.

### Separable spaces

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
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...

, most of the Hilbert spaces are in fact separable, as stipulated by the Wightman axioms
Wightman axioms
In physics the Wightman axioms are an attempt at a mathematically rigorous formulation of quantum field theory. Arthur Wightman formulated the axioms in the early 1950s but they were first published only in 1964, after Haag-Ruelle scattering theory affirmed their significance.The axioms exist in...

. However, it is sometimes argued that non-separable Hilbert spaces are also important in quantum field theory, roughly because the systems in the theory possess an infinite number of degrees of freedom and any infinite Hilbert tensor product (of spaces of dimension greater than one) is non-separable. For instance, a bosonic field
Bosonic field
In quantum field theory, a bosonic field is a quantum field whose quanta are bosons; that is, they obey Bose-Einstein statistics. Bosonic fields obey canonical commutation relations, as distinct from the canonical anticommutation relations obeyed by fermionic fields.Examples include scalar fields,...

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 S is a subset of a Hilbert space H, the set of vectors orthogonal to S is defined by
S is a closed
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...

subspace of H (can be proved easily using the linearity and continuity of the inner product) and so forms itself a Hilbert space. If V is a closed subspace of H, then V is called the orthogonal complement of V. In fact, every x in H can then be written uniquely as x = v + w, with v in V and w in V. Therefore, H is the internal Hilbert direct sum of V and V.

The linear operator PV : HH which maps x to v is called the orthogonal projection onto V. There is a natural
Natural transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed this intuition...

one-to-one correspondence between the set of all closed subspaces of H and the set of all bounded self-adjoint operators P such that P2 = P. Specifically,
Theorem. The orthogonal projection PV is a self-adjoint linear operator on H of norm ≤ 1 with the property P2V = PV. Moreover, any self-adjoint linear operator E such that E2 = E is of the form PV, where V is the range of E. For every x in H, PV(x) is the unique element v of V which minimizes the distance ||xv||.

This provides the geometrical interpretation of PV(x): it is the best approximation to x by elements of V.

An operator P such that P = P2 = P∗ is called an orthogonal projection. The orthogonal projection PV onto a closed subspace V of H is the adjoint of the inclusion mapping
meaning that
for all x ∈ H and y ∈ V. Projections PU and PV are called mutually orthogonal if PUPV = 0. This is equivalent to U and V being orthogonal as subspaces of H. As a result, the sum of the two projections PU and PV is only a projection if U and V are orthogonal to each other, and in that case PU + PV
= PU+V. The composite PUPV is generally not a projection; in fact, the composite is a projection if and only if the two projections commute, and in that case PUPV = PUV.

The operator norm of a projection P onto a non-zero closed subspace is equal to one:
Every closed subspace V of a Hilbert space is therefore the image of an operator P of norm one such that P2 = P. In fact this property characterizes Hilbert spaces:
• A Banach space of dimension higher than 2 is (isometrically) a Hilbert space if and only if, to every closed subspace V, there is an operator PV of norm one whose image is V such that

While this result characterizes the metric structure of a Hilbert space, the structure of a Hilbert space as a topological vector space
Topological vector space
In mathematics, a topological vector space is one of the basic structures investigated in functional analysis...

can itself be characterized in terms of the presence of complementary subspaces:
• A Banach space X is topologically and linearly isomorphic to a Hilbert space if and only if, to every closed subspace V, there is a closed subspace W such that X 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 V is contained in the closure
Closure (topology)
In mathematics, the closure of a subset S in a topological space consists of all points in S plus the limit points of S. Intuitively, these are all the points that are "near" S. A point which is in the closure of S is a point of closure of S...

of U. This result is a special case of the Hahn–Banach theorem
Hahn–Banach theorem
In mathematics, the Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed...

. The closure of a subspace can be completely characterized in terms of the orthogonal complement: If V is a subspace of H, then the closure of V is equal to . The orthogonal complement is thus a Galois connection
Galois connection
In mathematics, especially in order theory, a Galois connection is a particular correspondence between two partially ordered sets . The same notion can also be defined on preordered sets or classes; this article presents the common case of posets. Galois connections generalize the correspondence...

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: . If the Vi are in addition closed, then .

## Spectral theory

There is a well-developed spectral theory
Spectral theory
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of...

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

The spectrum of an operator T, denoted σ(T) is the set of complex numbers λ such that T − λ lacks a continuous inverse. If T is bounded, then the spectrum is always a compact set in the complex plane, and lies inside the disc If T is self-adjoint, then the spectrum is real. In fact, it is contained in the interval [m,M] where
Moreover, m and M are both actually contained within the spectrum.

The eigenspaces of an operator T are given by
Unlike with finite matrices, not every element of the spectrum of T must be an eigenvalue: the linear operator T − λ 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
Spectral theorem
In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides conditions under which an operator or a matrix can be diagonalized...

of a self-adjoint operator T takes a particularly simple form if, in addition, T is assumed to be a compact operator
Compact operator
In functional analysis, a branch of mathematics, a compact operator is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset of Y...

. The spectral theorem for compact self-adjoint operators states:
• A compact self-adjoint operator T has only countably (or finitely) many spectral values. The spectrum of T has no limit point
Limit point
In mathematics, a limit point of a set S in a topological space X is a point x in X that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contains a point of S other than x itself. Note that x does not have to be an element of S...

in the complex plane except possibly zero. The eigenspaces of T decompose H into an orthogonal direct sum:
Moreover, if Eλ denotes the orthogonal projection onto the eigenspace Hλ, then
where the sum converges with respect to the norm on B(H).

This theorem plays a fundamental role in the theory of integral equation
Integral equation
In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. There is a close connection between differential and integral equations, and some problems may be formulated either way...

s, 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 T associates to each real number λ an operator Eλ, which is the projection onto the nullspace of the operator , where the positive part of a self-adjoint operator is defined by
The operators Eλ 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
The integral is understood as a Riemann–Stieltjes integral, convergent with respect to the norm on B(H). In particular, one has the ordinary scalar-valued integral representation
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 dEλ 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 T any continuous complex function ƒ defined on the spectrum of T by forming the integral
The resulting continuous functional calculus
Continuous functional calculus
In mathematics, the continuous functional calculus of operator theory and C*-algebra theory allows applications of continuous functions to normal elements of a C*-algebra. More precisely,Theorem...

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: λ is a spectral value if the resolvent operator
fails to be a well-defined continuous operator. The self-adjointness of T still guarantees that the spectrum is real. Thus the essential idea of working with unbounded operators is to look instead at the resolvent Rλ where λ is non-real. This is a bounded normal operator, which admits a spectral representation that can then be transferred to a spectral representation of T 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
Riesz potential
In mathematics, the Riesz potential is a potential named after its discoverer, the Hungarian mathematician Marcel Riesz. In a sense, the Riesz potential defines an inverse for a power of the Laplace operator on Euclidean space...

or Bessel potential
Bessel potential
In mathematics, the Bessel potential is a potential similar to the Riesz potential but with better decay properties at infinity....

.

A precise version of the spectral theorem which holds in this case is:
Given a densely defined self-adjoint operator T on a Hilbert space H, there corresponds a unique resolution of the identity E on the Borel sets of R, such that
for all x ∈ D(T) and y ∈ H. The spectral measure E is concentrated on the spectrum of T.

There is also a version of the spectral theorem that applies to unbounded normal operators.

• Hilbert C*-module
• Hilbert algebra
• Hilbert manifold
Hilbert manifold
In mathematics, a Hilbert manifold is a manifold modeled on Hilbert spaces. Thus it is a separable Hausdorff space in which each point has a neighbourhood homeomorphic to an infinite dimensional Hilbert space. The concept of a Hilbert manifold provides a possibility of extending the theory of...

• Rigged Hilbert space
Rigged Hilbert space
In mathematics, a rigged Hilbert space is a construction designed to link the distribution and square-integrable aspects of functional analysis. Such spaces were introduced to study spectral theory in the broad sense...

• Topologies on the set of operators on a Hilbert space
• Operator theory
Operator theory
In mathematics, operator theory is the branch of functional analysis that focuses on bounded linear operators, but which includes closed operators and nonlinear operators.Operator theory also includes the study of algebras of operators....