Functional analysis

# Functional analysis

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Functional analysis is a branch of 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...

, the core of which is formed by the study of 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...

s endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear operator
Linear transformation
In mathematics, a linear map, linear mapping, linear transformation, or linear operator is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or 0...

s acting upon these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions
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:...

and the formulation of properties of transformations of functions such as 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...

as transformations defining 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...

, 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...

etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations.

The usage of the word functional
Functional (mathematics)
In mathematics, and particularly in functional analysis, a functional is a map from a vector space into its underlying scalar field. In other words, it is a function that takes a vector as its input argument, and returns a scalar...

goes back to 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...

, implying a function whose argument is a function and the name was first used in Hadamard's 1910 book on that subject. However, the general concept of functional had previously been introduced in 1887 by the Italian mathematician and physicist Vito Volterra
Vito Volterra
Vito Volterra was an Italian mathematician and physicist, known for his contributions to mathematical biology and integral equations....

. The theory of nonlinear functionals was continued by students of Hadamard, in particular Fréchet and Lévy. Hadamard also founded the modern school of linear functional analysis further developed by Riesz and the group
Lwów School of Mathematics
The Lwów School of Mathematics was a group of mathematicians who worked between the two World Wars in Lviv, then known as Lwów and located in Poland, but now located in western Ukraine. The mathematicians often met at the famous Scottish Café to discuss mathematical problems, and published in the...

of Polish
Poland
Poland , officially the Republic of Poland , is a country in Central Europe bordered by Germany to the west; the Czech Republic and Slovakia to the south; Ukraine, Belarus and Lithuania to the east; and the Baltic Sea and Kaliningrad Oblast, a Russian exclave, to the north...

mathematicians around Stefan Banach
Stefan Banach
Stefan Banach was a Polish mathematician who worked in interwar Poland and in Soviet Ukraine. He is generally considered to have been one of the 20th century's most important and influential mathematicians....

.

In modern introductory texts to functional analysis, the subject is seen as the study of vector spaces endowed with a topology, in particular infinite dimensional spaces
Dimension (vector space)
In mathematics, the dimension of a vector space V is the cardinality of a basis of V. It is sometimes called Hamel dimension or algebraic dimension to distinguish it from other types of dimension...

. In contrast, 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...

deals mostly with finite dimensional spaces, or does not use topology. An important part of functional analysis is the extension of the theory of measure
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...

, integration
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

, and probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

to infinite dimensional spaces, also known as infinite dimensional analysis.

## Normed vector spaces

The basic and historically first class of spaces studied in functional analysis are 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....

normed vector space
Normed vector space
In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any real vector space Rn. The following properties of "vector length" are crucial....

s over the 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 number
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...

s. Such spaces are called 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. An important example is a Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

, where the norm arises from an inner product. These spaces are of fundamental importance in many areas, including the mathematical formulation of 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...

.

More generally, functional analysis includes the study of Fréchet space
Fréchet space
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces...

s and other topological vector space
Topological vector space
In mathematics, a topological vector space is one of the basic structures investigated in functional analysis...

s not endowed with a norm.

An important object of study in functional analysis are the continuous linear operators
Linear transformation
In mathematics, a linear map, linear mapping, linear transformation, or linear operator is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or 0...

defined on Banach and Hilbert spaces. These lead naturally to the definition of C*-algebras and other 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.

### Hilbert spaces

Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

s can be completely classified: there is a unique Hilbert space up to
Up to
In mathematics, the phrase "up to x" means "disregarding a possible difference in  x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...

isomorphism
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations.  If there exists an isomorphism between two structures, the two structures are said to be isomorphic.  In a certain sense, isomorphic structures are...

for every 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...

of the 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...

. Finite-dimensional Hilbert spaces are fully understood in 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...

, and infinite-dimensional separable Hilbert spaces are isomorphic to . Separability being important for applications, functional analysis of Hilbert spaces consequently mostly deals with this space. One of the open problems in functional analysis is to prove that every bounded linear operator on a Hilbert space has a proper invariant subspace
Invariant subspace
In mathematics, an invariant subspace of a linear mappingfrom some vector space V to itself is a subspace W of V such that T is contained in W...

. Many special cases of this invariant subspace problem have already been proven.

### Banach spaces

General 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 are more complicated than Hilbert spaces, and cannot be classified in such a simple manner as those. In particular, Banach spaces lack a notion analogous to 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...

.

Examples of Banach spaces are -spaces
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...

for any real number . Given also a measure on set , then , sometimes also denoted or , has as its vectors equivalence classes of measurable functions whose 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...

's -th power has finite integral, that is, functions for which one has.

If is the counting measure
Counting measure
In mathematics, the counting measure is an intuitive way to put a measure on any set: the "size" of a subset is taken to be the number of elements in the subset, if the subset is finite, and ∞ if the subset is infinite....

, then the integral may be replaced by a sum. That is, we require.

Then it is not necessary to deal with equivalence classes, and the space is denoted , written more simply in the case when is the set of non-negative integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

s.

In Banach spaces, a large part of the study involves the dual space: the space of all continuous linear maps from the space into its underlying field, so-called functionals. A Banach space can be canonically identified with a subspace of its bidual, which is the dual of its dual space. The corresponding map 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...

but in general not onto. A general Banach space and its bidual need not even be isometrically isomorphic in any way, contrary to the finite-dimensional situation. This is explained in the dual space article.

Also, the notion of derivative
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...

can be extended to arbitrary functions between Banach spaces. See, for instance, the Fréchet derivative
Fréchet derivative
In mathematics, the Fréchet derivative is a derivative defined on Banach spaces. Named after Maurice Fréchet, it is commonly used to formalize the concept of the functional derivative used widely in the calculus of variations. Intuitively, it generalizes the idea of linear approximation from...

article.

## Major and foundational results

Important results of functional analysis include:
• 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...

(also known as Banach–Steinhaus theorem) applies to sets of operators with uniform bounds.
• One of 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...

s (there are indeed more than one) gives an integral formula for the normal operators on a Hilbert space. This theorem is of central importance for the mathematical formulation of 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...

.
• 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...

extends functionals from a subspace to the full space, in a norm-preserving fashion. An implication is the non-triviality of dual 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...

and 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 :...

.

See also: List of functional analysis topics.

## Foundations of mathematics considerations

Most spaces considered in functional analysis have infinite dimension. To show the existence of a vector space basis for such spaces may require 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...

. However, a somewhat different concept, Schauder basis
Schauder basis
In mathematics, a Schauder basis or countable basis is similar to the usual basis of a vector space; the difference is that Hamel bases use linear combinations that are finite sums, while for Schauder bases they may be infinite sums...

, is usually more relevant in functional analysis. Many very important theorems require 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...

, usually proved using axiom of choice, although the strictly weaker Boolean prime ideal theorem
Boolean prime ideal theorem
In mathematics, a prime ideal theorem guarantees the existence of certain types of subsets in a given abstract algebra. A common example is the Boolean prime ideal theorem, which states that ideals in a Boolean algebra can be extended to prime ideals. A variation of this statement for filters on...

suffices. The Baire category theorem
Baire category theorem
The Baire category theorem is an important tool in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space....

, needed to prove many important theorems, also requires a form of axiom of choice.

## Points of view

Functional analysis in its includes the following tendencies:
• Abstract analysis. An approach to analysis based on topological group
Topological group
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...

s, topological ring
Topological ring
In mathematics, a topological ring is a ring R which is also a topological space such that both the addition and the multiplication are continuous as mapswhere R × R carries the product topology.- General comments :...

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

s;
• Geometry of 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
contains many topics. One is combinatorial approach connected with Jean Bourgain
Jean Bourgain
Jean Bourgain is a Belgian mathematician. He has been a faculty member at the University of Illinois, Urbana-Champaign and, from 1985 until 1995, professor at Institut des Hautes Études Scientifiques at Bures-sur-Yvette in France, and since 1994 at the Institute for Advanced Study in Princeton,...

; another is a characterization of Banach spaces in which various forms of the law of large numbers
Law of large numbers
In probability theory, the law of large numbers is a theorem that describes the result of performing the same experiment a large number of times...

hold.
• Noncommutative geometry
Noncommutative geometry
Noncommutative geometry is a branch of mathematics concerned with geometric approach to noncommutative algebras, and with construction of spaces which are locally presented by noncommutative algebras of functions...

. Developed by Alain Connes
Alain Connes
Alain Connes is a French mathematician, currently Professor at the Collège de France, IHÉS, The Ohio State University and Vanderbilt University.-Work:...

, partly building on earlier notions, such as George Mackey
George Mackey
George Whitelaw Mackey was an American mathematician. Mackey earned his bachelor of arts at Rice University in 1938 and obtained his Ph.D. at Harvard University in 1942 under the direction of Marshall H. Stone...

's approach to 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....

.
• Connection with 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...

. Either narrowly defined as in mathematical physics
Mathematical physics
Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...

, or broadly interpreted by, e.g. 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...

, to include most types of 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...

.