Functional analysis is a branch of

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

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

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 ...

.

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

''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

Topics in Real and Functional Analysis

by Gerald Teschl, University of Vienna.

Lecture Notes on Functional Analysis

by Yevgeny Vilensky, New York University.

Lecture videos on functional analysis

b

Greg Morrow

from University of Colorado Colorado Springs {{Authority control Functional analysis,

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 completenormed 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 $\backslash ell^(\backslash aleph\_0)\backslash ,$. 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

GeneralBanach 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 $\backslash mu$ on set then sometimes also denoted $L^p(X,\backslash mu)$ or has as its vectors equivalence classes $;\; href="/html/ALL/s/,f\backslash ,.html"\; ;"title=",f\backslash ,">,f\backslash ,$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
:$\backslash int\_\backslash left,\; f(x)\backslash ^p\backslash ,d\backslash mu(x)\; <\; +\backslash infty.$
If $\backslash mu$ 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
:$\backslash sum\_\backslash left,\; f(x)\backslash ^p<+\backslash infty\; .$
Then it is not necessary to deal with equivalence classes, and the space is denoted written more simply $\backslash ell^p$ 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.
Linear functional analysis

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, theClosed 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

Theuniform 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 anormed 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 :$\backslash sup\backslash nolimits\_\; \backslash ,\; T(x)\backslash ,\; \_Y\; <\; \backslash infty,$ then :$\backslash sup\backslash nolimits\_\; \backslash ,\; T\backslash ,\; \_\; <\; \backslash infty.$

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 ameasure 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
:$U^*\; T\; U\; =\; A\; \backslash ;$
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 ...

:
:$;\; href="/html/ALL/s/\_\backslash varphi.html"\; ;"title="\; \backslash varphi">\; \backslash varphi$
and $\backslash ,\; T\backslash ,\; =\; \backslash ,\; f\backslash ,\; \_\backslash infty$
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

TheHahn–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" 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,
:$\backslash varphi(x)\; \backslash leq\; p(x)\backslash qquad\backslash forall\; x\; \backslash in\; U$
then there exists a linear extension of to the whole space ; that is, there exists a linear functional such that
:$\backslash psi(x)=\backslash varphi(x)\backslash qquad\backslash forall\; x\backslash in\; U,$
:$\backslash psi(x)\; \backslash le\; p(x)\backslash qquad\backslash forall\; x\backslash in\; V.$
Open mapping theorem

The open mapping theorem, also known as the Banach–Schauder theorem (named afterStefan 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 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 atopological 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 Other topics

Foundations of mathematics considerations

Most spaces considered in functional analysis have infinite dimension. To show the existence of avector 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 ontopological 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 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), ...

.
See also

* List of functional analysis topics * Spectral theoryReferences

Further reading

* Aliprantis, C.D., Border, K.C.: ''Infinite Dimensional Analysis: A Hitchhiker's Guide'', 3rd ed., Springer 2007, . Online (by subscription) * Bachman, G., Narici, L.: ''Functional analysis'', Academic Press, 1966. (reprint Dover Publications) * Stefan Banach, Banach S.]''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

External links

*Topics in Real and Functional Analysis

by Gerald Teschl, University of Vienna.

Lecture Notes on Functional Analysis

by Yevgeny Vilensky, New York University.

Lecture videos on functional analysis

b

Greg Morrow

from University of Colorado Colorado Springs {{Authority control Functional analysis,