Functional analysis: Difference between revisions

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[[Image:Drum vibration mode12.gif|thumb|right|200px|One of the possible modes of vibration of an idealized circular [[drum head]]. 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#Definition|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]]s 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=live| website=acsu.buffalo.edu | publisher=Proceedings of the May 1997 Meeting in Perugia}}</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 [[Maurice René 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]].