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Functional analysis

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Area of mathematics

This article is about an area of mathematics. For a method of study of human behavior, seeFunctional analysis (psychology). For a method in linguistics, seeFunctional analysis (linguistics).

One of the possible modes ofvibration of a circular membrane. These modes areeigenfunctions of a linear operator on a function space, a common construction in functional analysis.

Functional analysis is a branch ofmathematical analysis, the core of which is formed by the study ofvector spaces endowed with some kind of limit-related structure (for example,inner product,norm, ortopology) and thelinear functions defined on these spaces and suitably respecting these structures. The historical roots of functional analysis lie in the study ofspaces of functions and the formulation of properties of transformations of functions such as theFourier transform as transformations defining, for example,continuous orunitary operators between function spaces. This point of view turned out to be particularly useful for the study ofdifferential andintegral equations.

The usage of the wordfunctional as a noun goes back to thecalculus of variations, implying afunction whose argument is a function. The term was first used inHadamard's 1910 book on that subject. However, the general concept of a functional had previously been introduced in 1887 by the Italian mathematician and physicistVito Volterra.^[1]^[2] The theory of nonlinear functionals was continued by students of Hadamard, in particularFréchet andLévy. Hadamard also founded the modern school of linear functional analysis further developed byRiesz and thegroup ofPolish mathematicians aroundStefan Banach.

In modern introductory texts on functional analysis, the subject is seen as the study of vector spaces endowed with a topology, in particularinfinite-dimensional spaces.^[3]^[4] In contrast,linear algebra deals mostly with finite-dimensional spaces, and does not use topology. An important part of functional analysis is the extension of the theories ofmeasure,integration, andprobability to infinite-dimensional spaces, also known asinfinite dimensional analysis.

Normed vector spaces

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The basic and historically first class of spaces studied in functional analysis arecomplete normed vector spaces over thereal orcomplex numbers. Such spaces are calledBanach spaces. An important example is aHilbert space, where the norm arises from an inner product. These spaces are of fundamental importance in many areas, including themathematical formulation of quantum mechanics,machine learning,partial differential equations, andFourier analysis.

More generally, functional analysis includes the study ofFréchet spaces and othertopological vector spaces not endowed with a norm.

An important object of study in functional analysis are thecontinuous linear operators defined on Banach and Hilbert spaces. These lead naturally to the definition ofC*-algebras and otheroperator algebras.

Hilbert spaces

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Hilbert spaces can be completely classified: there is a unique Hilbert spaceup to isomorphism for everycardinality of theorthonormal basis.^[5] Finite-dimensional Hilbert spaces are fully understood inlinear algebra, and infinite-dimensionalseparable Hilbert spaces are isomorphic to $\ell ^{\,2}(\aleph _{0})\,$ . 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 properinvariant subspace. Many special cases of thisinvariant subspace problem have already been proven.

Banach spaces

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GeneralBanach spaces 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 anorthonormal basis.

Examples of Banach spaces are $L^{p}$ -spaces for any real number $p\geq 1$ . Given also a measure $\mu$ on set $X {\displaystyle X}$ , then $L^{p}(X)$ , sometimes also denoted $L^{p}(X,\mu )$ or $L^{p}(\mu )$ , has as its vectors equivalence classes $[\,f\,]$ ofmeasurable functions whoseabsolute value's $p {\displaystyle p}$ -th power has finite integral; that is, functions $f {\displaystyle f}$ for which one has $\int _{X}\left|f(x)\right|^{p}\,d\mu (x)<\infty .$

If $\mu$ is thecounting measure, then the integral may be replaced by a sum. That is, we require $\sum _{x\in X}\left|f(x)\right|^{p}<\infty .$

Then it is not necessary to deal with equivalence classes, and the space is denoted $\ell ^{p}(X)$ , written more simply $\ell ^{p}$ in the case when $X {\displaystyle X}$ is the set of non-negativeintegers.

In Banach spaces, a large part of the study involves thedual space: the space of allcontinuous 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 anisometry 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 ofderivative can be extended to arbitrary functions between Banach spaces. See, for instance, theFréchet derivative article.

Linear functional analysis

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^[6]

This sectionneeds expansion. You can help byadding to it.(August 2020)

Major and foundational results

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There are four major theorems which are sometimes called the four pillars of functional analysis:

theHahn–Banach theorem
theopen mapping theorem
theclosed graph theorem
theuniform boundedness principle, also known as theBanach–Steinhaus theorem.

Important results of functional analysis include:

Uniform boundedness principle

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Main article:Banach-Steinhaus theorem

Theuniform boundedness principle orBanach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with theHahn–Banach theorem and theopen mapping theorem, it is considered one of the cornerstones of the field. In its basic form, it asserts that for a family ofcontinuous linear operators (and thus bounded operators) whose domain is aBanach space, pointwise boundedness is equivalent to uniform boundedness in operator norm.

The theorem was first published in 1927 byStefan Banach andHugo Steinhaus but it was also proven independently byHans Hahn.

Theorem (Uniform Boundedness Principle)—Let $X {\displaystyle X}$ be aBanach space and $Y {\displaystyle Y}$ be anormed vector space. Suppose that $F {\displaystyle F}$ is a collection of continuous linear operators from $X {\displaystyle X}$ to $Y {\displaystyle Y}$ . If for all $x {\displaystyle x}$ in $X {\displaystyle X}$ one has $\sup \nolimits _{T\in F}\|T(x)\|_{Y}<\infty ,$ then $\sup \nolimits _{T\in F}\|T\|_{B(X,Y)}<\infty .$

Spectral theorem

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Main article:Spectral theorem

There are many theorems known as thespectral theorem, but one in particular has many applications in functional analysis.

Spectral theorem^[7]—Let $A {\displaystyle A}$ be a bounded self-adjoint operator on a Hilbert space $H {\displaystyle H}$ . Then there is ameasure space $(X,\Sigma ,\mu )$ and a real-valuedessentially bounded measurable function $f {\displaystyle f}$ on $X {\displaystyle X}$ and a unitary operator $U:H\to L_{\mu }^{2}(X)$ such that $U^{*}TU=A$ whereT is themultiplication operator: $[T\varphi ](x)=f(x)\varphi (x).$ and $\|T\|=\|f\|_{\infty }$ .

This is the beginning of the vast research area of functional analysis calledoperator theory; see also thespectral measure.

There is also an analogous spectral theorem for boundednormal operators on Hilbert spaces. The only difference in the conclusion is that now $f {\displaystyle f}$ may be complex-valued.

Hahn–Banach theorem

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Main article:Hahn–Banach theorem

TheHahn–Banach theorem is a central tool in functional analysis. It allows the extension ofbounded linear functionals defined on a subspace of somevector space to the whole space, and it also shows that there are "enough"continuous linear functionals defined on everynormed vector space to make the study of thedual space "interesting".

Hahn–Banach theorem:^[8]—If $p:V\to \mathbb {R}$ is asublinear function, and $\varphi :U\to \mathbb {R}$ is alinear functional on alinear subspace $U\subseteq V$ which is dominated by $p {\displaystyle p}$ on $U {\displaystyle U}$ ; that is, $\varphi (x)\leq p(x)\qquad \forall x\in U$ then there exists a linear extension $\psi :V\to \mathbb {R}$ of $\varphi$ to the whole space $V {\displaystyle V}$ which is dominated by $p {\displaystyle p}$ on $V {\displaystyle V}$ ; that is, there exists a linear functional $\psi$ such that ${\begin{aligned}\psi (x)&=\varphi (x)&\forall x\in U,\\\psi (x)&\leq p(x)&\forall x\in V.\end{aligned}}$

Open mapping theorem

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Main article:Open mapping theorem (functional analysis)

Theopen mapping theorem, also known as the Banach–Schauder theorem (named afterStefan Banach andJuliusz Schauder), is a fundamental result which states that if acontinuous linear operator betweenBanach spaces issurjective then it is anopen map. More precisely,^[8]

Open mapping theorem—If $X {\displaystyle X}$ and $Y {\displaystyle Y}$ are Banach spaces and $A:X\to Y$ is a surjective continuous linear operator, then $A {\displaystyle A}$ is an open map (that is, if $U {\displaystyle U}$ is anopen set in $X {\displaystyle X}$ , then $A(U)$ is open in $Y {\displaystyle Y}$ ).

The proof uses theBaire category theorem, and completeness of both $X {\displaystyle X}$ and $Y {\displaystyle Y}$ is essential to the theorem. The statement of the theorem is no longer true if either space is just assumed to be anormed space, but is true if $X {\displaystyle X}$ and $Y {\displaystyle Y}$ are taken to beFréchet spaces.

Closed graph theorem

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Main article:Closed graph theorem

Closed graph theorem—If $X {\displaystyle X}$ is atopological space and $Y {\displaystyle Y}$ is acompact Hausdorff space, then the graph of a linear map $T {\displaystyle T}$ from $X {\displaystyle X}$ to $Y {\displaystyle Y}$ is closed if and only if $T {\displaystyle T}$ iscontinuous.^[9]

Foundations of mathematics considerations

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Most spaces considered in functional analysis have infinite dimension. To show the existence of avector space basis for such spaces may requireZorn's lemma. However, a somewhat different concept, theSchauder basis, is usually more relevant in functional analysis. Many theorems require theHahn–Banach theorem, usually proved using theaxiom of choice, although the strictly weakerBoolean prime ideal theorem suffices. TheBaire category theorem, needed to prove many important theorems, also requires a form of axiom of choice.

Points of view

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Functional analysis includes the following tendencies:

Abstract analysis. An approach to analysis based ontopological groups,topological rings, andtopological vector spaces.
Geometry ofBanach spaces contains many topics. One iscombinatorial approach connected withJean Bourgain; another is a characterization of Banach spaces in which various forms of thelaw of large numbers hold.
Noncommutative geometry. Developed byAlain Connes, partly building on earlier notions, such asGeorge Mackey's approach toergodic theory.
Connection withquantum mechanics. Either narrowly defined as inmathematical physics, or broadly interpreted by, for example,Israel Gelfand, to include most types ofrepresentation theory.

References

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^Lawvere, F. William."Volterra's functionals and covariant cohesion of space"(PDF).acsu.buffalo.edu. Proceedings of the May 1997 Meeting in Perugia. Archived fromthe original(PDF) on 2003-04-07. Retrieved2018-06-12.
^Saraiva, Luís (October 2004).History of Mathematical Sciences. WORLD SCIENTIFIC. p. 195.doi:10.1142/5685.ISBN 978-93-86279-16-3.
^Bowers, Adam; Kalton, Nigel J. (2014).An introductory course in functional analysis.Springer. p. 1.
^Kadets, Vladimir (2018).A Course in Functional Analysis and Measure Theory [КУРС ФУНКЦИОНАЛЬНОГО АНАЛИЗА].Springer. pp. xvi.
^Riesz, Frigyes (1990).Functional analysis. Béla Szőkefalvi-Nagy, Leo F. Boron (Dover ed.). New York: Dover Publications. pp. 195–199.ISBN 0-486-66289-6.OCLC 21228994.
^Rynne, Bryan; Youngson, Martin A. (29 December 2007).Linear Functional Analysis. Springer. RetrievedDecember 30, 2023.
^Hall, Brian C. (2013-06-19).Quantum Theory for Mathematicians.Springer Science & Business Media. p. 147.ISBN 978-1-4614-7116-5.
^^a ^bRudin, Walter (1991).Functional Analysis. McGraw-Hill.ISBN 978-0-07-054236-5.
^Munkres, James R. (2000).Topology. Prentice Hall, Incorporated. p. 171.ISBN 978-0-13-181629-9.

External links

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"Functional analysis",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
Topics in Real and Functional Analysis byGerald Teschl, University of Vienna.
Lecture Notes on Functional Analysis by Yevgeny Vilensky, New York University.
Lecture videos on functional analysis byGreg Morrow Archived 2017-04-01 at theWayback Machine fromUniversity of Colorado Colorado Springs

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