- 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 mathematician and physicist Vito Volterra and its founding is largely attributed to mathematician Stefan Banach.
Normed vector spaces
In the modern view, functional analysis is seen as the study of complete normed vector spaces
over the real
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.
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
) 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.
General 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).
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
. Many very important theorems require the Hahn-Banach theorem
, usually proved using axiom of choice
, although the strictly weaker Boolean prime ideal theorem
Points of view
Functional analysis in its present form
includes the following tendencies:
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- 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
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- 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