Editing Functional analysis

{{short description|Area of mathematics}}
{{about|an area of mathematics|a method of study of human behavior|Functional analysis (psychology)|a method in linguistics|Functional analysis (linguistics)}}

[[Image:Drum vibration mode12.gif|thumb|right|One of the possible modes of [[vibration of a circular membrane]]. These modes are [[eigenfunction]]s of a linear operator on a function space, a common construction in functional analysis.]]

'''Functional analysis''' is a branch of [[mathematical analysis]], the core of which is formed by the study of [[vector space]]s endowed with some kind of limit-related structure (for example, [[Inner product space#Definition|inner product]], [[Norm (mathematics)#Definition|norm]], or [[Topological space#Definitions|topology]]) and the [[linear transformation|linear function]]s defined on these spaces and suitably respecting these structures. The historical roots of functional analysis lie in the study of [[function space|spaces of functions]] and the formulation of properties of transformations of functions such as the [[Fourier transform]] as transformations defining, for example, [[continuous function|continuous]] or [[unitary operator|unitary]] operators between function spaces. This point of view turned out to be particularly useful for the study of [[differential equations|differential]] and [[integral equations]].

The usage of the word ''[[functional (mathematics)|functional]]'' as a noun goes back to the [[calculus of variations]], implying a [[Higher-order function|function whose argument is a function]]. The term was first used in [[Jacques Hadamard|Hadamard]]'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]].<ref>{{Cite web|last=Lawvere|first=F. William|title=Volterra's functionals and covariant cohesion of space|url=http://www.acsu.buffalo.edu/~wlawvere/Volterra.pdf|archive-url=https://web.archive.org/web/20030407030553/http://www.acsu.buffalo.edu/~wlawvere/Volterra.pdf|archive-date=2003-04-07|url-status=dead|website=acsu.buffalo.edu|publisher=Proceedings of the May 1997 Meeting in Perugia|access-date=2018-06-12}}</ref><ref>{{Cite book| url=http://dx.doi.org/10.1142/5685|title=History of Mathematical Sciences|date=October 2004| page=195| publisher=WORLD SCIENTIFIC| doi=10.1142/5685|isbn=978-93-86279-16-3|last1=Saraiva|first1=Luís}}</ref> The theory of nonlinear functionals was continued by students of Hadamard, in particular [[René Maurice Fréchet|Fréchet]] and [[Paul Lévy (mathematician)|Lévy]]. Hadamard also founded the modern school of linear functional analysis further developed by [[Frigyes Riesz|Riesz]] and the [[Lwów School of Mathematics|group]] of [[Poland|Polish]] mathematicians around [[Stefan Banach]].

In modern introductory texts on functional analysis, the subject is seen as the study of vector spaces endowed with a topology, in particular [[Dimension (vector space)|infinite-dimensional spaces]].<ref>{{Cite book| last1=Bowers|first1=Adam|title=An introductory course in functional analysis|last2=Kalton|first2=Nigel J.| publisher=[[Springer Science & Business Media|Springer]]|year=2014|pages=1}}</ref><ref>{{Cite book| last=Kadets| first=Vladimir| title=A Course in Functional Analysis and Measure Theory|publisher=[[Springer Publishing|Springer]] | year=2018|pages=xvi|trans-title=КУРС ФУНКЦИОНАЛЬНОГО АНАЛИЗА}}</ref> 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 of [[measure (mathematics)|measure]], [[integral|integration]], and [[probability]] 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 space|complete]] [[normed vector space]]s over the [[real number|real]] or [[complex number]]s. Such spaces are called [[Banach space]]s. An important example is a [[Hilbert space]], where the norm arises from an inner product. These spaces are of fundamental importance in many areas, including the [[mathematical formulation of quantum mechanics]], [[Reproducing kernel Hilbert space|machine learning]], [[partial differential equations]], and [[Fourier analysis]].

More generally, functional analysis includes the study of [[Fréchet space]]s and other [[topological vector space]]s not endowed with a norm.

An important object of study in functional analysis are the [[continuous function (topology)|continuous]] [[linear transformation|linear operators]] defined on Banach and Hilbert spaces. These lead naturally to the definition of [[C*-algebra]]s and other [[operator algebra]]s.

===Hilbert spaces===
[[Hilbert space]]s can be completely classified: there is a unique Hilbert space [[up to]] [[isomorphism]] for every [[cardinal number|cardinality]] of the [[orthonormal basis]].<ref>{{Cite book| last=Riesz|first=Frigyes| url=https://www.worldcat.org/oclc/21228994|title=Functional analysis|date=1990|publisher=Dover Publications| others = Béla Szőkefalvi-Nagy, Leo F. Boron|isbn=0-486-66289-6|edition=Dover |location=New York|oclc=21228994| pages = 195–199}}</ref> Finite-dimensional Hilbert spaces are fully understood in [[linear algebra]], and infinite-dimensional [[Separable space|separable]] Hilbert spaces are isomorphic to [[Sequence space#ℓp spaces|<math>\ell^{\,2}(\aleph_0)\,</math>]]. Separability being important for applications, functional analysis of Hilbert spaces consequently mostly deals with this space. One of the open problems in functional analysis is to prove that every bounded linear operator on a Hilbert space has a proper [[invariant subspace]]. Many special cases of this [[invariant subspace problem]] have already been proven.

===Banach spaces===
General [[Banach space]]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]].

Examples of Banach spaces are [[Lp space|<math>L^p</math>-spaces]] for any real number {{nowrap|<math>p\geq1</math>.}} Given also a measure <math>\mu</math> on set {{nowrap|<math>X</math>,}} then {{nowrap|<math>L^p(X)</math>,}} sometimes also denoted <math>L^p(X,\mu)</math> or {{nowrap|<math>L^p(\mu)</math>,}} has as its vectors equivalence classes <math>[\,f\,]</math> of [[Lebesgue-measurable function|measurable function]]s whose [[absolute value]]'s <math>p</math>-th power has finite integral; that is, functions <math>f</math> for which one has
<math display="block">\int_{X}\left|f(x)\right|^p\,d\mu(x) < \infty.</math>

If <math>\mu</math> is the [[counting measure]], then the integral may be replaced by a sum. That is, we require
<math display="block">\sum_{x\in X}\left|f(x)\right|^p < \infty .</math>

Then it is not necessary to deal with equivalence classes, and the space is denoted {{nowrap|<math>\ell^p(X)</math>,}} written more simply <math>\ell^p</math> in the case when <math>X</math> is the set of non-negative [[integer]]s.

In Banach spaces, a large part of the study involves the [[Continuous dual|dual space]]: the space of all [[continuous function (topology)|continuous]] linear maps from the space into its underlying field, so-called functionals. A Banach space can be canonically identified with a subspace of its bidual, which is the dual of its dual space. The corresponding map is an [[isometry]] 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]] can be extended to arbitrary functions between Banach spaces. See, for instance, the [[Fréchet derivative]] article.

==Linear functional analysis==
<ref>{{Cite book|url=https://www.amazon.com/Functional-Analysis-Springer-Undergraduate-Mathematics-ebook/dp/B00FBSNUCQ/ref=sr_1_1?crid=1U7ZU6A5UTQY3&keywords=linear+functional+analysis&qid=1703979184&s=digital-text&sprefix=linear+functional+analysis%2Cdigital-text%2C152&sr=1-1|title=Linear Functional Analysis|first1=Bryan|last1=Rynne|first2=Martin A.|last2=Youngson|date=29 December 2007 |publisher=Springer |access-date=December 30, 2023}}</ref>
{{expand section|date=August 2020}}

==Major and foundational results==

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

* the [[Hahn–Banach theorem]]
* the [[Open mapping theorem (functional analysis)|open mapping theorem]]
* the [[Closed graph theorem (functional analysis)|closed graph theorem]]
* the [[uniform boundedness principle]], also known as the [[Banach–Steinhaus theorem]]. 

Important results of functional analysis include:

===Uniform boundedness principle===
{{main|Banach-Steinhaus theorem}}
The [[uniform boundedness principle]] or [[Banach–Steinhaus theorem]] is one of the fundamental results in functional analysis. Together with the [[Hahn–Banach theorem]] and the [[open mapping theorem (functional analysis)|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 operator]]s (and thus bounded operators) whose domain is a [[Banach space]], pointwise boundedness is equivalent to uniform boundedness in operator norm.

The theorem was first published in 1927 by [[Stefan Banach]] and [[Hugo Steinhaus]] but it was also proven independently by [[Hans Hahn (mathematician)|Hans Hahn]].

{{math theorem | name = Theorem (Uniform Boundedness Principle) | math_statement = Let <math>X</math> be a [[Banach space]] and <math>Y</math> be a [[normed vector space]]. Suppose that <math>F</math> is a collection of continuous linear operators from <math>X</math> to <math>Y</math>. If for all <math>x</math> in <math>X</math> one has
<math display="block">\sup\nolimits_{T \in F} \|T(x)\|_Y  < \infty, </math>
then
<math display="block">\sup\nolimits_{T \in F} \|T\|_{B(X,Y)}  < \infty.</math>}}

===Spectral theorem===
{{main|Spectral theorem}}
There are many theorems known as the [[spectral theorem]], but one in particular has many applications in functional analysis.

{{math theorem | name = Spectral theorem<ref>{{Cite book|last=Hall|first=Brian C.|url={{google books |plainurl=y |id=bYJDAAAAQBAJ|page=147}}|title=Quantum Theory for Mathematicians|date=2013-06-19|publisher=[[Springer Science & Business Media]]|isbn=978-1-4614-7116-5|page=147|language=en}}</ref>
|math_statement = Let <math>A</math>  be a bounded self-adjoint operator on a Hilbert space <math>H</math>. Then there is a [[measure space]] <math>(X,\Sigma,\mu)</math> and a real-valued [[ess sup|essentially bounded]] measurable function <math>f</math> on <math>X</math> and a unitary operator <math>U:H\to L^2_\mu(X)</math> such that
<math display="block"> U^* T U = A </math>
where ''T'' is the [[multiplication operator]]:
<math display="block"> [T \varphi](x) = f(x) \varphi(x). </math>
and <math>\|T\| = \|f\|_\infty</math>.}}

This is the beginning of the vast research area of functional analysis called [[operator theory]]; see also the [[spectral measure#Spectral measure|spectral measure]].

There is also an analogous spectral theorem for bounded [[normal operator]]s on Hilbert spaces. The only difference in the conclusion is that now <math>f</math> may be complex-valued.

===Hahn–Banach theorem===
{{main|Hahn–Banach theorem}}
The [[Hahn–Banach theorem]] is a central tool in functional analysis. It allows the extension of [[Bounded operator|bounded linear functionals]] defined on a subspace of some [[vector space]] to the whole space, and it also shows that there are "enough" [[continuous function (topology)|continuous]] linear functionals defined on every [[normed vector space]] to make the study of the [[dual space]] "interesting".

{{math theorem | name = Hahn–Banach theorem:<ref name="rudin">{{Cite book | last=Rudin | first=Walter | url={{google books |plainurl=y |id=Sh_vAAAAMAAJ}} | title=Functional Analysis | date=1991 | publisher=McGraw-Hill | isbn=978-0-07-054236-5 | language=en}}</ref>
| math_statement = If <math>p:V\to\mathbb{R}</math> is a [[sublinear function]], and <math>\varphi:U\to\mathbb{R}</math> is a [[linear functional]] on a [[linear subspace]] <math>U\subseteq V</math> which is dominated by <math>p</math> on <math>U</math>; that is,
<math display="block">\varphi(x) \leq p(x)\qquad\forall x \in U</math>
then there exists a linear extension <math>\psi:V\to\mathbb{R}</math> of <math>\varphi</math> to the whole space <math>V</math> which is dominated by <math>p</math> on <math>V</math>; that is, there exists a linear functional <math>\psi</math> such that
<math display="block">\begin{align}
\psi(x) &= \varphi(x) &\forall x\in U, \\
\psi(x) &\le p(x) &\forall x\in V.
\end{align}</math>}}

===Open mapping theorem===
{{main|Open mapping theorem (functional analysis)}}
The [[open mapping theorem (functional analysis)|open mapping theorem]], also known as the Banach–Schauder theorem (named after [[Stefan Banach]] and [[Juliusz Schauder]]), is a fundamental result which states that if a [[Bounded linear operator|continuous linear operator]] between [[Banach space]]s is [[surjective]] then it is an [[open map]]. More precisely,<ref name=rudin/>
{{math theorem | name = Open mapping theorem | math_statement = If <math>X</math> and <math>Y</math> are Banach spaces and <math>A:X\to Y</math> is a surjective continuous linear operator, then <math>A</math> is an open map (that is, if <math>U</math> is an [[open set]] in <math>X</math>, then <math>A(U)</math> is open in <math>Y</math>).}}

The proof uses the [[Baire category theorem]], and completeness of both <math>X</math> and <math>Y</math> 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]], but is true if <math>X</math> and <math>Y</math> are taken to be [[Fréchet space]]s.

===Closed graph theorem===
{{main|Closed graph theorem}}
{{math theorem | name = Closed graph theorem | math_statement = If <math>X</math> is a [[topological space]] and <math>Y</math> is a [[Compact space|compact]] [[Hausdorff space]], then the graph of a linear map <math>T</math> from <math>X</math> to <math>Y</math> is closed if and only if <math>T</math> is [[continuous function (topology)|continuous]].<ref>{{Cite book | last=Munkres | first=James R. | url={{google books |plainurl=y |id=XjoZAQAAIAAJ}} | title=Topology | date=2000 | publisher=Prentice Hall, Incorporated | isbn=978-0-13-181629-9 | language=en | page= 171}}</ref>}}

===Other topics===
{{main|List of functional analysis topics}}

==Foundations of mathematics considerations==
Most spaces considered in functional analysis have infinite dimension. To show the existence of a [[vector space basis]] for such spaces may require [[Zorn's lemma]]. However, a somewhat different concept, the [[Schauder basis]], is usually more relevant in functional analysis. Many theorems require the [[Hahn–Banach theorem]], usually proved using the [[axiom of choice]], although the strictly weaker [[Boolean prime ideal theorem]] suffices. The [[Baire category theorem]], needed to prove many important theorems, also requires a form of axiom of choice.

==Points of view==
Functional analysis includes the following tendencies:
*''Abstract analysis''. An approach to analysis based on [[topological group]]s, [[topological ring]]s, and [[topological vector space]]s.
*''Geometry of [[Banach space]]s'' contains many topics. One is [[combinatorial]] approach connected with [[Jean Bourgain]]; another is a characterization of Banach spaces in which various forms of the [[law of large numbers]] hold.
*''[[Noncommutative geometry]]''. Developed by [[Alain Connes]], partly building on earlier notions, such as [[George Mackey]]'s approach to [[ergodic theory]].
*''Connection with [[quantum mechanics]]''. Either narrowly defined as in [[mathematical physics]], or broadly interpreted by, for example, [[Israel Gelfand]], to include most types of [[representation theory]].

==See also==

* [[List of functional analysis topics]]
* [[Spectral theory]]

==References==
{{Reflist}}

==Further reading==
* Aliprantis, C.D., Border, K.C.: ''Infinite Dimensional Analysis: A Hitchhiker's Guide'', 3rd ed., Springer 2007, {{ISBN|978-3-540-32696-0}}. Online {{doi|10.1007/3-540-29587-9}} (by subscription)
* Bachman, G., Narici, L.: ''Functional analysis'', Academic Press, 1966. (reprint Dover Publications)
* [[Stefan Banach|Banach S.]] [http://www.ebook3000.com/Theory-of-Linear-Operations--Volume-38--North-Holland-Mathematical-Library--by-S--Banach_134628.html ''Theory of Linear Operations''] {{Webarchive|url=https://web.archive.org/web/20211028084954/http://www.ebook3000.com/Theory-of-Linear-Operations--Volume-38--North-Holland-Mathematical-Library--by-S--Banach_134628.html |date=2021-10-28 }}. Volume 38, North-Holland Mathematical Library, 1987, {{ISBN|0-444-70184-2}}
* [[Haïm Brezis|Brezis, H.]]: ''Analyse Fonctionnelle'', Dunod {{ISBN|978-2-10-004314-9}} or {{ISBN|978-2-10-049336-4}}
* [[John B. Conway|Conway, J. B.]]: ''A Course in Functional Analysis'', 2nd edition, Springer-Verlag, 1994, {{ISBN|0-387-97245-5}}
* [[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, {{ISBN|0-444-51790-1}}
* 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, {{ISBN|0-471-55604-1}}
* 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, {{ISBN|978-0-8176-4367-6}}
* [[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.
* [[Sergei 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 ==
{{Sister project links| wikt=no | commons=no | b=Functional Analysis | n=no | q=Functional analysis | s=no | v=no | voy=no | species=no | d=no}}

* {{Springer |title=Functional analysis |id=p/f042020}}
* [https://www.mat.univie.ac.at/~gerald/ftp/book-fa/index.html Topics in Real and Functional Analysis] by [[Gerald Teschl]], University of Vienna.
* [https://web.archive.org/web/20161021093450/http://www.math.nyu.edu/phd_students/vilensky/Functional_Analysis.pdf Lecture Notes on Functional Analysis] by Yevgeny Vilensky, New York University.
* [https://www.youtube.com/playlist?list=PLE1C83D79C93E2266 Lecture videos on functional analysis] by [http://www.uccs.edu/~gmorrow/ Greg Morrow] {{Webarchive|url=https://web.archive.org/web/20170401143239/http://www.uccs.edu/~gmorrow/ |date=2017-04-01 }} from [[University of Colorado Colorado Springs]]

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