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Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier transform, as well as 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 Italianmarker mathematician and physicist Vito Volterra and its founding is largely attributed to a group of Polishmarker mathematicians around Stefan Banach. In the modern view, functional analysis is seen as the study of vector spaces endowed with a topology, in particular infinitely dimensional spaces. In contrast, linear algebra deals mostly with finite dimensional spaces, or does not use topology. An important part of functional analysis is the extension of the theory of measure, integration, and probability to infinite dimensional spaces, also known as infinitely dimensional analysis.

Normed vector spaces

The basic and historically first class of spaces studied in functional analysis are 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.

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 and other operator algebras.

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 (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 bounded linear operator on a Hilbert space has a proper invariant subspace. Many special cases of this invariant subspace problem have already been proven.

Banach spaces

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

Examples of Banach spaces are L^{p}-spaces, for any real number p\geq1 (see Lp spaces). Given p\geq1 and a set X (which may be countable or uncountable), L^{p}(X) consists of "all Lebesgue-measurable functions whose absolute value's p-th power has finite integral".

That is, it consists of all Lebesgue-measurable functions f for which \int_{x\in X}\left|f(x)\right|^p\,dx<+\INFTY<></+\INFTY<>math>.

If X is countable, the integral may be replaced with a sum: \sum_{X}\left|f(x)\right|^p<+\INFTY<></+\INFTY<>math>, although for countable X, the space is usually denoted l^p(X).

In Banach spaces, a large part of the study involves the dual space: the space of all continuous linear functionals. The dual of the dual is not always isomorphic to the original space, but there is always a natural monomorphism from a space into its dual's dual. This is explained in the dual space article.

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

Major and foundational results

Important results of functional analysis include:

See also: List of functional analysis topics.

Foundations of mathematics considerations

Most 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. However, a somewhat different concept, Schauder basis, is usually more relevant in functional analysis. Many very important theorems require the Hahn-Banach theorem, usually proved using axiom of choice, although the strictly weaker Boolean prime ideal theorem suffices. The Baire category theorem, needed to prove many important theorems, also requires a form of axiom of choice.

Points of view

Functional analysis in its includes the following tendencies:

References

  • Brezis, H.: Analyse Fonctionnelle, Dunod ISBN 978-2100043149 or ISBN 978-2100493364
  • Conway, John B.: A Course in Functional Analysis, 2nd edition, Springer-Verlag, 1994, ISBN 0-387-97245-5
  • Dunford, N. and Schwartz, J.T. : Linear Operators, General Theory, and other 3 volumes, includes visualization charts
  • Eidelman, Yuli, Vitali Milman, and Antonis Tsolomitis: Functional Analysis: An Introduction, American Mathematical Society, 2004.
  • Giles,J.R.: Introduction to the Analysis of Normed Linear Spaces,Cambridge University Press,2000
  • Hirsch F., Lacombe G. - "Elements of Functional Analysis", Springer 1999.
  • Hutson, V., Pym, J.S., Cloud M.J.: Applications of Functional Analysis and Operator Theory, 2nd edition, Elsevier Science, 2005, ISBN 0-444-51790-1
  • Kolmogorov, A.N and Fomin, S.V.: Elements of the Theory of Functions and Functional Analysis, Dover Publications, 1999
  • Kreyszig, Erwin: Introductory Functional Analysis with Applications, Wiley, 1989.
  • Lax, P.: Functional Analysis, Wiley-Interscience, 2002
  • Lebedev, L.P. and Vorovich, I.I.: Functional Analysis in Mechanics, Springer-Verlag, 2002
  • Michel, Anthony N. and Charles J. Herget: Applied Algebra and Functional Analysis, Dover, 1993.
  • Reed M., Simon B. - "Functional Analysis", Academic Press 1980.
  • Riesz, F. and Sz.-Nagy, B.: Functional Analysis, Dover Publications, 1990
  • Rudin, W.: Functional Analysis, McGraw-Hill Science, 1991
  • Schechter, M.: Principles of Functional Analysis, AMS, 2nd edition, 2001
  • Shilov, Georgi E.: Elementary Functional Analysis, Dover, 1996.
  • Sobolev, S.L.: Applications of Functional Analysis in Mathematical Physics, AMS, 1963
  • Yosida, K.: Functional Analysis, Springer-Verlag, 6th edition, 1980


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