# functional analysis

Functional analysis   is the branch of mathematics, and specifically of analysis  , concerned with the study of spaces of functions. It has its historical roots in the study of transformations  , such as the Fourier transform, and in the study of differential  and integral equations. This usage of the word functional    goes back to the calculus of variations, implying a function whose argument   is a function. Its use in general has been attributed to mathematician and physicist Vito Volterra and its founding is largely attributed to mathematician Stefan Banach.

## 0.1 Normed vector spaces

An important object of study in functional analysis are the continuous linear operators defined on Banach and Hilbert spaces. These lead naturally to the definition of $C^{*}$-algebras   (http://planetmath.org/CAlgebra) and other operator algebras.

## 0.2 Hilbert spaces

Hilbert 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 ($\aleph_{0}$) 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 the invariant subspace problem, which conjectures that every operator on a Hilbert space has a non-trivial invariant subspace  . Many special cases have already been proven.

## 0.3 Banach spaces

General Banach spaces are more complicated. There is no clear definition of what would constitute a base, for example.

## 0.4 Major and foundational results

Important results of functional analysis include:

## 0.6 Points of view

Functional analysis in its present form includes the following tendencies:

Soft analysis. An approach to analysis based on topological groups, topological rings, and topological vector spaces; Geometry of Banach spaces. A combinatorial approach primarily due to Jean Bourgain; Noncommutative geometry. Developed by Alain Connes, partly building on earlier notions, such as George Mackey’s approach to ergodic theory; Connection with quantum mechanics. Either narrowly defined as in mathematical physics, or broadly interpreted by, e.g. Israel Gelfand, to include most types of representation theory.

This entry was adapted from the Wikipedia article http://en.wikipedia.org/wiki/Functional_analysisFunctional analysis as of December 18, 2006.

Title functional analysis FunctionalAnalysis 2013-03-22 16:28:19 2013-03-22 16:28:19 PrimeFan (13766) PrimeFan (13766) 12 PrimeFan (13766) Topic msc 26E35