- For functional analysis as used in psychology, see the
functional analysis
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
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
(ℵ
_{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 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
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<></+\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
External links