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Normed vector spacesIn the modern view, functional analysis is seen as the study of complete normed vector spaces over the real or complex numbers. Such spaces are called Banach spaces. 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. More generally, functional analysis includes the study of Fréchet spaces and other topological vector spaces not endowed with a norm.
Hilbert spacesHilbert spaces can be completely classified: there is a unique Hilbert space up to isomorphism for every cardinality of the base. Since finite-dimensional Hilbert spaces are fully understood in linear algebra, and since morphisms of Hilbert spaces can always be divided into morphisms of spaces with Aleph-null (ℵ0) dimensionality, functional analysis of Hilbert spaces mostly deals with the unique Hilbert space of dimensionality Aleph-null, and its morphisms. One of the open problems in functional analysis is to prove that every operator on a Hilbert space has a proper invariant subspace. Many special cases have already been proven. Banach spacesGeneral Banach spaces are more complicated. There is no clear definition of what would constitute a base, for example. For any real number p ≥ 1, an example of a Banach space is given by "all Lebesgue-measurable functions whose absolute value's p-th power has finite integral" (see Lp spaces).
Also, the notion of derivative can be extended to arbitrary functions between Banach spaces. See, for instance, the Fréchet derivative article. Major and foundational resultsImportant results of functional analysis include:
See also: List of functional analysis topics. Foundations of mathematics considerationsMost 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. Many very important theorems require the Hahn-Banach theorem, which relies on the axiom of choice that is strictly weaker than the Boolean prime ideal theorem. Points of viewFunctional analysis in its present form includes the following tendencies:
References
de:Funktionalanalysis es:Análisis funcional fa:آنالیز تابعی fr:Analyse fonctionnelle (mathématiques) it:Analisi funzionale he:אנליזה פונקציונלית ka:ფუნქციონალური ანალიზი nl:Functionaalanalyse ja:関数解析学 pl:Analiza funkcjonalna pt:Análise funcional ru:Функциональный анализ fi:Funktionaalianalyysi sv:Funktionalanalys uk:Функціональний аналіз zh:泛函分析
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