Editing Functional analysis (section)

{{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'''.