Content deleted Content added
update link |
Rescuing 2 sources and tagging 0 as dead.) #IABot (v2.0.9.5 |
||
Line 6:
'''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]] 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
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]]|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'''.
Line 125:
* 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}}
| |||