Functional analysis is a branch of mathematical analysis, the core of which is formed by the
study of vector spaces endowed with some kind of limit-related structure and the linear
operators acting upon these spaces and respecting these structures in a suitable sense. The
historical roots of functional analysis lie in the study of spaces of functions and the
formulation of properties of transformations of functions such as the Fourier transform
as transformations defining continuous, unitary etc. operators between function spaces. This
point of view turned out to be particularly useful for the study of differential and integral
equations. The usage of the word functional goes back
to the calculus of variations, implying a function whose argument is a function and
the name was first used in Hadamard's 1910 book on that subject. However, the general
concept of a functional had previously been introduced in 1887 by the Italian mathematician
and physicist Vito Volterra. The theory of nonlinear functionals was continued by students
of Hadamard, in particular Fréchet and Lévy. Hadamard also founded the modern school of
linear functional analysis further developed by Riesz and the group of Polish mathematicians
around Stefan Banach. In modern introductory texts to functional
analysis, the subject 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, and 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.
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.
Finite-dimensional Hilbert spaces are fully understood in linear algebra, and infinite-dimensional
separable Hilbert spaces are isomorphic to . Separability being important for applications,
functional analysis of Hilbert spaces consequently mostly deals with this space. 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 than Hilbert spaces, and cannot be classified
in such a simple manner as those. In particular, many Banach spaces lack a notion analogous
to an orthonormal basis. Examples of Banach spaces are -spaces for
any real number . Given also a measure on set , then , sometimes also denoted or , has
as its vectors equivalence classes of measurable functions whose absolute value's -th power
has finite integral, that is, functions for which one has
. If is the counting measure, then the integral
may be replaced by a sum. That is, we require .
Then it is not necessary to deal with equivalence classes, and the space is denoted , written
more simply in the case when is the set of non-negative integers.
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 applies to sets of operators with uniform bounds.
One of the spectral theorems 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 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.
See also List of functional analysis topics
Spectral theory References
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Academic Press, 1966. Banach S. Theory of Linear Operations. Volume
38, North-Holland Mathematical Library, 1987, ISBN 0-444-70184-2
Brezis, H.: Analyse Fonctionnelle, Dunod ISBN 978-2-10-004314-9 or ISBN 978-2-10-049336-4
Conway, J. 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 Edwards, R. E.: Functional Analysis, Theory
and Applications, Hold, Rinehart and Winston, 1965.
Eidelman, Yuli, Vitali Milman, and Antonis Tsolomitis: Functional Analysis: An Introduction,
American Mathematical Society, 2004. Freidman, A.: Foundations of Modern Analysis,
Dover Publications, Paperback Edition, July 21, 2010
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, E.: 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 978-0-8176-4367-6
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
Vogt, D., Meise, R.: Introduction to Functional Analysis, Oxford University Press, 1997.
External links Hazewinkel, Michiel, ed., "Functional analysis",
Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 Topics in Real and 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.
Lecture videos on functional analysis by Greg Morrow from University of Colorado Colorado
Springs An Introduction to Functional Analysis on
Coursera by John Cagnol from Ecole Centrale Paris