TheInfoList

Functional analysis is a branch of
mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathematics), series, and analytic ...
, the core of which is formed by the study of
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s endowed with some kind of limit-related structure (e.g.
inner product In mathematics, an inner product space or a Hausdorff space, Hausdorff pre-Hilbert space is a vector space with a binary operation called an inner product. This operation associates each pair of vectors in the space with a Scalar (mathematics), ...

,
norm Norm, the Norm or NORM may refer to: In academic disciplines * Norm (geology), an estimate of the idealised mineral content of a rock * Norm (philosophy) Norms are concepts ( sentences) of practical import, oriented to effecting an action, rat ...

,
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

, etc.) and the
linear function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
s defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the
Fourier transform#REDIRECT Fourier transform In mathematics, a Fourier transform (FT) is a Integral transform, mathematical transform that decomposes function (mathematics), functions depending on space or time into functions depending on spatial or temporal frequenc ...
as transformations defining
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ga ...
,
unitary Unitary may refer to: * Unitary construction, in automotive design a common term for unibody (unitary body/chassis) construction * Lethal Unitary Chemical Agents and Munitions (Unitary), as chemical weapons opposite of Binary * Unitarianism, in Chr ...
etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and
integral equations In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. The usage of the word ''
functional Functional may refer to: * Movements in architecture: ** Functionalism (architecture) In architecture File:Plan d'exécution du second étage de l'hôtel de Brionne (dessin) De Cotte 2503c – Gallica 2011 (adjusted).jpg, upright=1.45, alt=Pl ...
'' as a noun goes back to the
calculus of variations The calculus of variations is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of functionals: Map (mathematic ...
, implying a function whose argument is a function. The term was first used in
Hadamard Jacques Salomon Hadamard ForMemRS (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis of the function . Hue represents the argument, brightness the magnitud ...
's 1910 book on that subject. However, the general concept of a functional had previously been introduced in 1887 by the Italian mathematician and physicist
Vito Volterra Vito Volterra (, ; 3 May 1860 – 11 October 1940) was an Italian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as q ...

. 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 A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
of
Polish Polish may refer to: * Anything from or related to Poland Poland ( pl, Polska ), officially the Republic of Poland ( pl, Rzeczpospolita Polska, links=no ), is a country located in Central Europe. It is divided into 16 Voivodeships of Pol ...

mathematicians around
Stefan Banach Stefan Banach ( ; 30 March 1892 – 31 August 1945) was a Polish mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quan ...

. 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. In contrast,
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mat ...
deals mostly with finite-dimensional spaces, and does not use topology. An important part of functional analysis is the extension of the theory of measure, , and
probability Probability is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...

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
normed vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s over the
real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
or
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

s. Such spaces are called
Banach space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s. An important example is a
Hilbert space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
, where the norm arises from an inner product. These spaces are of fundamental importance in many areas, including the
mathematical formulation of quantum mechanics The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, ab ...
,
machine learning Machine learning (ML) is the study of computer algorithms that can improve automatically through experience and by the use of data. It is seen as a part of artificial intelligence. Machine learning algorithms build a model based on sample data ...
,
partial differential equations In , a partial differential equation (PDE) is an equation which imposes relations between the various s of a . The function is often thought of as an "unknown" to be solved for, similarly to how is thought of as an unknown number to be solved ...
, and
Fourier analysis In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
. More generally, functional analysis includes the study of
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 (normed vector spaces that are Complete space, complete with ...
s and other
topological vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s not endowed with a norm. An important object of study in functional analysis are the
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ga ...
linear operators In mathematics, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \rightarrow W between two vector spaces that preserves the operat ...
defined on Banach and Hilbert spaces. These lead naturally to the definition of
C*-algebra In mathematics, specifically in functional analysis Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a commo ...
s and other
operator algebra In functional analysis Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional ...
s.

## Hilbert spaces

Hilbert space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
s can be completely classified: there is a unique Hilbert space
up to Two mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
isomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

for every
cardinality In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of the
orthonormal basis In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
. Finite-dimensional Hilbert spaces are fully understood in
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mat ...
, and infinite-dimensional separable Hilbert spaces are isomorphic to $\ell^\left(\aleph_0\right)\,$. 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 subspaceIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
. Many special cases of this
invariant subspace problem In the field of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, a ...
have already been proven.

## Banach spaces

General
Banach space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s are more complicated than Hilbert spaces, and cannot be classified in such a simple manner as those. In particular, many Banach spaces lack a notion analogous to an
orthonormal basis In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
. Examples of Banach spaces are $L^p$-spaces for any real number Given also a measure $\mu$ on set then sometimes also denoted $L^p\left(X,\mu\right)$ or has as its vectors equivalence classes
measurable function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s whose
absolute value In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

's $p$-th power has finite integral; that is, functions $f$ for which one has :$\int_\left, f\left(x\right)\^p\,d\mu\left(x\right) < +\infty.$ If $\mu$ is the
counting measure In mathematics, specifically measure theory, the counting measure is an intuitive way to put a Measure (mathematics), measure on any Set (mathematics), set – the "size" of a subset is taken to be the number of elements in the subset if the subset ...
, then the integral may be replaced by a sum. That is, we require :$\sum_\left, f\left(x\right)\^p<+\infty .$ Then it is not necessary to deal with equivalence classes, and the space is denoted written more simply $\ell^p$ in the case when $X$ is the set of non-negative
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
s. In Banach spaces, a large part of the study involves the
dual space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
: the space of all
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ga ...
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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

can be extended to arbitrary functions between Banach spaces. See, for instance, the
Fréchet derivative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
article.

# Major and foundational results

There are four major theorems which are sometimes called the four pillars of functional analysis. These are the Hahn-Banach theorem, the Open Mapping theorem, the
Closed Graph theorem In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
and the
Uniform Boundedness Principle In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
also known as the Banach-Steinhaus theorem. Important results of functional analysis include:

## Uniform boundedness principle

The
uniform boundedness principle In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the
Hahn–Banach theorem The Hahn–Banach theorem is a central tool in functional analysis Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function ...
and the open mapping theorem, it is considered one of the cornerstones of the field. In its basic form, it asserts that for a family of
continuous linear operatorIn functional analysis and related areas of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, chang ...
s (and thus bounded operators) whose domain is a
Banach space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, pointwise boundedness is equivalent to uniform boundedness in operator norm. The theorem was first published in 1927 by
Stefan Banach Stefan Banach ( ; 30 March 1892 – 31 August 1945) was a Polish mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quan ...

and
Hugo Steinhaus Władysław Hugo Dionizy Steinhaus (January 14, 1887 – February 25, 1972) was a Jewish-Polish mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes t ...

but it was also proven independently by Hans Hahn.
Theorem (Uniform Boundedness Principle). Let ''X'' be a
Banach space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
and ''Y'' be a
normed vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
. Suppose that ''F'' is a collection of continuous linear operators from ''X'' to ''Y''. If for all ''x'' in ''X'' one has :$\sup\nolimits_ \, T\left(x\right)\, _Y < \infty,$ then :$\sup\nolimits_ \, T\, _ < \infty.$

## Spectral theorem

There are many theorems known as the spectral theorem, but one in particular has many applications in functional analysis. Theorem: Let ''A'' be a bounded self-adjoint operator on a Hilbert space ''H''. Then there is a
measure space A measure space is a basic object of measure theory Measure is a fundamental concept of mathematics. Measures provide a mathematical abstraction for common notions like mass, distance/length, area, volume, probability of events, and — after si ...
and a real-valued essentially bounded measurable function ''f'' on ''X'' and a unitary operator such that :$U^* T U = A \;$ where ''T'' is the
multiplication operator In operator theoryIn mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded ...
: : and $\, T\, = \, f\, _\infty$ This is the beginning of the vast research area of functional analysis called
operator theoryIn mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators or ...
; see also the spectral measure. There is also an analogous spectral theorem for bounded
normal operator In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s on Hilbert spaces. The only difference in the conclusion is that now $f$ may be complex-valued.

## Hahn–Banach theorem

The
Hahn–Banach theorem The Hahn–Banach theorem is a central tool in functional analysis Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function ...
is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
to the whole space, and it also shows that there are "enough"
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ga ...
linear functionals defined on every
normed vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
to make the study of the
dual space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
"interesting". Hahn–Banach theorem: If is a sublinear function, and is a
linear functional In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
on a
linear subspace In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
which is dominated by on ; that is, :$\varphi\left(x\right) \leq p\left(x\right)\qquad\forall x \in U$ then there exists a linear extension of to the whole space ; that is, there exists a linear functional such that :$\psi\left(x\right)=\varphi\left(x\right)\qquad\forall x\in U,$ :$\psi\left(x\right) \le p\left(x\right)\qquad\forall x\in V.$

## Open mapping theorem

The open mapping theorem, also known as the Banach–Schauder theorem (named after
Stefan Banach Stefan Banach ( ; 30 March 1892 – 31 August 1945) was a Polish mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quan ...

and
Juliusz Schauder Juliusz Paweł Schauder (; 21 September 1899, Lwów Lviv ( uk, Львів ; orv, Львівград; pl, Lwów ; yi, לעמבערג, Lemberg; russian: Львов, Lvov ; german: Lemberg; la, Leopolis; hu, Ilyvó; see also other names) is ...
), is a fundamental result which states that if a
continuous linear operatorIn functional analysis and related areas of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, chang ...
between
Banach space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s is
surjective In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
then it is an
open map In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
. More precisely,: : Open mapping theorem. If ''X'' and ''Y'' are Banach spaces and ''A'' : ''X'' → ''Y'' is a surjective continuous linear operator, then ''A'' is an open map (that is, if ''U'' is an
open set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
in ''X'', then ''A''(''U'') is open in ''Y''). The proof uses the
Baire category theorem The Baire category theorem (BCT) is an important result in general topology , a useful example in point-set topology. It is connected but not path-connected. In mathematics, general topology is the branch of topology that deals with the basic Set t ...
, and completeness of both ''X'' and ''Y'' is essential to the theorem. The statement of the theorem is no longer true if either space is just assumed to be a
normed space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
, but is true if ''X'' and ''Y'' are taken to be
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 (normed vector spaces that are Complete space, complete with ...
s.

## Closed graph theorem

The closed graph theorem states the following: If ''X'' is a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
and ''Y'' is a
compact Compact as used in politics may refer broadly to a pact A pact, from Latin ''pactum'' ("something agreed upon"), is a formal agreement. In international relations International relations (IR), international affairs (IA) or internationa ...
Hausdorff space In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), ...

, then the graph of a linear map ''T'' from ''X'' to ''Y'' is closed if and only if ''T'' is
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ga ...
.

# Foundations of mathematics considerations

Most spaces considered in functional analysis have infinite dimension. To show the existence of a
vector space basis In mathematics, a Set (mathematics), set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred ...
for such spaces may require
Zorn's lemma Zorn's lemma, also known as the Kuratowski–Zorn lemma, after mathematicians Max August Zorn, Max Zorn and Kazimierz Kuratowski, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (ord ...
. However, a somewhat different concept,
Schauder basisIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
, is usually more relevant in functional analysis. Many very important theorems require the
Hahn–Banach theorem The Hahn–Banach theorem is a central tool in functional analysis Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function ...
, usually proved using the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

, although the strictly weaker
Boolean prime ideal theorem In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
suffices. The
Baire category theorem The Baire category theorem (BCT) is an important result in general topology , a useful example in point-set topology. It is connected but not path-connected. In mathematics, general topology is the branch of topology that deals with the basic Set t ...
, 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 In mathematics, topological groups are logically the combination of Group (mathematics), groups and Topological space, topological spaces, i.e. they are groups and topological spaces at the same time, such that the Continuous function, continui ...
s,
topological ringIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
s, and
topological vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s. *''Geometry of
Banach space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s'' contains many topics. One is
combinatorial Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite set, finite Mathematical structure, structures. It is closely related to many other are ...
approach connected with
Jean Bourgain Jean, Baron Bourgain (; – ) was a Belgian Belgians ( nl, Belgen, french: Belges, german: Belgier) are people identified with the Kingdom of Belgium, a federal state in Western Europe. As Belgium is a multinational state, this connection ...

; another is a characterization of Banach spaces in which various forms of the
law of large numbers In probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...

hold. *''
Noncommutative geometryNoncommutative geometry (NCG) is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, chang ...
''. Developed by
Alain Connes Alain Connes (; born 1 April 1947) is a French mathematician, and a theoretical physicist, known for his contributions to the study of operator algebras and noncommutative geometryNoncommutative geometry (NCG) is a branch of mathematics ...

, partly building on earlier notions, such as
George Mackey George Whitelaw Mackey (February 1, 1916 – March 15, 2006) was an American mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such to ...
's approach to
ergodic theory Ergodic theory ( Greek: ' "work", ' "way") is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
. *''Connection with
quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...
''. Either narrowly defined as in
mathematical physics Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developme ...
, or broadly interpreted by, for example,
Israel Gelfand Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gel'fand, or Izrail M. Gelfand ( yi, ישראל געלפֿאַנד, russian: Изра́иль Моисе́евич Гельфа́нд; – 5 October 2009) was a prominent USSR, Soviet m ...
, to include most types of
representation theory Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...
.

* List of functional analysis topics * Spectral theory

# References

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''Theory of Linear Operations''
Volume 38, North-Holland Mathematical Library, 1987, * Haïm Brezis, Brezis, H.: ''Analyse Fonctionnelle'', Dunod or * John B. Conway, Conway, J. B.: ''A Course in Functional Analysis'', 2nd edition, Springer-Verlag, 1994, * Nelson Dunford, Dunford, N. and Jacob T. Schwartz, Schwartz, J.T.: ''Linear Operators, General Theory, John Wiley & Sons'', and other 3 volumes, includes visualization charts * Edwards, R. E.: ''Functional Analysis, Theory and Applications'', Hold, Rinehart and Winston, 1965. * Eidelman, Yuli, Vitali Milman, and Antonis Tsolomitis: ''Functional Analysis: An Introduction'', American Mathematical Society, 2004. * Avner Friedman, Friedman, A.: ''Foundations of Modern Analysis'', Dover Publications, Paperback Edition, July 21, 2010 * Giles,J.R.: ''Introduction to the Analysis of Normed Linear Spaces'',Cambridge University Press,2000 * Hirsch F., Lacombe G. - "Elements of Functional Analysis", Springer 1999. * Hutson, V., Pym, J.S., Cloud M.J.: ''Applications of Functional Analysis and Operator Theory'', 2nd edition, Elsevier Science, 2005, * Kantorovitz, S.,''Introduction to Modern Analysis'', Oxford University Press,2003,2nd ed.2006. * Kolmogorov, Kolmogorov, A.N and Sergei Fomin, Fomin, S.V.: ''Elements of the Theory of Functions and Functional Analysis'', Dover Publications, 1999 * Erwin Kreyszig, Kreyszig, E.: ''Introductory Functional Analysis with Applications'', Wiley, 1989. * Peter Lax, Lax, P.: ''Functional Analysis'', Wiley-Interscience, 2002, * Lebedev, L.P. and Vorovich, I.I.: ''Functional Analysis in Mechanics'', Springer-Verlag, 2002 * Michel, Anthony N. and Charles J. Herget: ''Applied Algebra and Functional Analysis'', Dover, 1993. * Pietsch, Albrecht: ''History of Banach spaces and linear operators'', Birkhäuser Boston Inc., 2007, * Michael C. Reed, Reed, M., Barry Simon, Simon, B.: "Functional Analysis", Academic Press 1980. * Riesz, F. and Sz.-Nagy, B.: ''Functional Analysis'', Dover Publications, 1990 * Walter Rudin, Rudin, W.: ''Functional Analysis'', McGraw-Hill Science, 1991 * Saxe, Karen: ''Beginning Functional Analysis'', Springer, 2001 * Schechter, M.: ''Principles of Functional Analysis'', AMS, 2nd edition, 2001 * Shilov, Georgi E.: ''Elementary Functional Analysis'', Dover, 1996. * Sobolev, Sobolev, S.L.: ''Applications of Functional Analysis in Mathematical Physics'', AMS, 1963 * Vogt, D., Meise, R.: ''Introduction to Functional Analysis'', Oxford University Press, 1997. * Kōsaku Yosida, Yosida, K.: ''Functional Analysis'', Springer-Verlag, 6th edition, 1980