Functional analysis

For functional analysis as used in psychology, see the functional analysis (psychology) article.

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 Italian mathematician and physicist Vito Volterra and its founding is largely attributed to a group of Polish mathematicians around Stefan Banach. In the modern view, functional analysis is seen as the study of vector spaces endowed with a topology, in particular infinite 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 infinite dimensional analysis.

1 Normed vector spaces
- 1.1 Hilbert spaces
- 1.2 Banach spaces
2 Major and foundational results
3 Foundations of mathematics considerations
4 Points of view
5 References
6 External links
7 See also

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 orthonormal basis. 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, and cannot be classified in such a simple manner as Hilbert spaces. In particular, they lack a notion analogous to an orthonormal basis.

Examples of Banach spaces are $L^{p}$ -spaces, for any real number $p\geq1$ (see L^p 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 $\int_{x\in X}\left|f(x)\right|^p\,dx<+\infty$ .

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

In Banach spaces, a large part of the study involves the dual space: the space of all continuous linear maps from the space into its underlying field, so-called functionals. A Banach space can be canonically identified with a subspace of its bidual, which is the dual of its dual space. The corresponding map is an isometry but in general not onto. A general Banach space and its bidual need not even be isometrically isomorphic in any way, contrary to the finite-dimensional situation. 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:

The uniform boundedness principle (also known as Banach–Steinhaus theorem) applies to sets of operators with uniform bounds.
One of the spectral theorems (there are indeed more than one) gives an integral formula for the normal operators on a Hilbert space. This theorem is of central importance for the mathematical formulation of quantum mechanics.
The Hahn–Banach theorem extends functionals from a subspace to the full space, in a norm-preserving fashion. An implication is the non-triviality of dual spaces.
The open mapping theorem and closed graph theorem.

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 present form^[update] includes the following tendencies:

Abstract analysis. An approach to analysis based on topological groups, topological rings, and topological vector spaces;
Geometry of Banach spaces contains many topics. One is combinatorial approach connected with Jean Bourgain; another is a characterization of Banach spaces in which various forms of the law of large numbers hold.
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.

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
Kantorovitz, S.,Introduction to Modern Analysis, Oxford University Press,2003,2nd ed.2006.
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.
Pietsch, Albrecht: History of Banach spaces and linear operators, Birkhauser Boston Inc., 2007, ISBN 978-0817643676
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

External links

Functional Analysis by Gerald Teschl, University of Vienna.
Lecture Notes on Functional Analysis by Yevgeny Vilensky, New York University.
Earliest Known Uses of Some of the Words of Mathematics: Calculus & Analysis by John Aldrich University of Southampton.