functional analysis

Functional analysisMathworldPlanetmathPlanetmath is the branch of mathematics, and specifically of analysisMathworldPlanetmath, concerned with the study of spaces of functions. It has its historical roots in the study of transformationsPlanetmathPlanetmath, such as the Fourier transform, and in the study of differentialMathworldPlanetmath and integral equations. This usage of the word functionalMathworldPlanetmathPlanetmathPlanetmath goes back to the calculus of variations, implying a function whose argumentMathworldPlanetmathPlanetmath 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

In the modern view, functional analysis is seen as the study of completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath normed vector spacesPlanetmathPlanetmath over the real or complex numbersMathworldPlanetmathPlanetmath. Such spaces are called Banach spacesMathworldPlanetmath. An important example is a Hilbert spaceMathworldPlanetmath, where the norm arises from an inner productMathworldPlanetmath. 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 spacesMathworldPlanetmath not endowed with a norm.

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*-algebrasMathworldPlanetmathPlanetmath ( and other operator algebras.

0.2 Hilbert spaces

Hilbert spaces can be completely classified: there is a unique Hilbert space up to isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath for every cardinality of the base. Since finite-dimensional Hilbert spaces are fully understood in linear algebraMathworldPlanetmath, and since morphismsMathworldPlanetmath of Hilbert spaces can always be divided into morphisms of spaces with Aleph-null (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 problemsPlanetmathPlanetmath in functional analysis is the invariant subspace problem, which conjectures that every operator on a Hilbert space has a non-trivial invariant subspacePlanetmathPlanetmath. 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.

For any real number p1, an example of a Banach space is given by ”all Lebesgue-measurable functions whose absolute valueMathworldPlanetmathPlanetmathPlanetmathPlanetmath’s p-th power has finite integralPlanetmathPlanetmath” (see Lp spaces).

In Banach spaces, a large part of the study involves the dual spaceMathworldPlanetmathPlanetmath: the space of all continuousMathworldPlanetmathPlanetmath linear functionalsMathworldPlanetmath. The dual of the dual is not always isomorphic to the original space, but there is always a natural monomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmath from a space into its dual’s dual. This is explained in the dual space article.

Also, the notion of derivativePlanetmathPlanetmath can be extended to arbitrary functions between Banach spaces. See, for instance, the Fréchet derivative.

0.4 Major and foundational results

Important results of functional analysis include:

The uniform boundedness principleMathworldPlanetmath applies to sets of operators with tight bounds. One of the spectral theoremsMathworldPlanetmathPlanetmath (there are indeed more than one) gives an integral formulaMathworldPlanetmathPlanetmath for the normal operators on a Hilbert space. This theoremMathworldPlanetmath is of central importance for the mathematical formulation of quantum mechanics.

The Hahn-Banach theoremMathworldPlanetmath extends functionals from a subspaceMathworldPlanetmathPlanetmath to the full space, in a norm-preserving fashion. An implicationMathworldPlanetmath is the non-triviality of dual spaces. The open mapping theoremMathworldPlanetmath and closed graph theorem. See also: List of functional analysis topics.

0.5 Foundations of mathematics considerations

Most spaces considered in functional analysis have infiniteMathworldPlanetmathPlanetmath dimensionMathworldPlanetmathPlanetmath. 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 choiceMathworldPlanetmath that is strictly weaker than the Boolean prime ideal theorem.

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 analysis as of December 18, 2006.

Title functional analysis
Canonical name FunctionalAnalysis
Date of creation 2013-03-22 16:28:19
Last modified on 2013-03-22 16:28:19
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 12
Author PrimeFan (13766)
Entry type Topic
Classification msc 26E35