Functional analysis: Difference between revisions

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{{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|200px|One of the possible modes of [[vibration of an idealizeda circular [[drum headmembrane]]. 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#DefinitionDefinitions|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=deviateddead|website=acsu.buffalo.edu|publisher=Proceedings of the May 1997 Meeting in Perugia|access-date=2018-06-12|archivedate=2003-04-07|archiveurl=https://web.archive.org/web/20030407030553/http://www.acsu.buffalo.edu/~wlawvere/Volterra.pdf}}</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==

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===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===

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==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}}

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===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>}}

The closed graph theorem states the following:

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===

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==Points of view==

Functional analysis in its {{As of|2004|alt=present form}} 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.