Functional analysis
Functional analysis is the branch of
mathematics, and specifically of
analysis, concerned with the study of spaces of
functions. It has its historical roots in the study of
transformations, such as the
Fourier transform, and 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.
In the modern view, functional analysis is seen as the study of
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 operator on a Hilbert space has a proper
invariant subspace. Many special cases 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.
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.
The notion of
derivative is extended to arbitrary functions between Banach spaces. It turns out that the derivative of a function at a certain point is really a continuous linear map.
Important results of functional analysis include:
*The
uniform boundedness principle applies to sets of operators with tight 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.
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, which relies on the
axiom of choice that is strictly weaker than the
Boolean prime ideal theorem.
Functional analysis in its
present form includes the following tendencies:
Soft analysis. An approach to analysis based on
topological groups,
topological rings, and
topological vector spaces;
Geometry of Banach spaces. A
combinatorial approach primarily due to
Jean Bourgain;
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.
* Yosida, K.:
Functional Analysis, Springer-Verlag, 6th edition, 1980
* Schechter, M.:
Principles of Functional Analysis, AMS, 2nd edition, 2001
* Hutson, V., Pym, J.S., Cloud M.J.:
Applications of Functional Analysis and Operator Theory, 2nd edition, Elsevier Science, 2005, ISBN 0444517901
* Dunford, N. and Schwartz, J.T. :
Linear Operators, General Theory, and other 3 volumes, includes visualization charts
* Brezis, H.:
Analyse Fonctionnelle, Dunod
* Sobolev, S.L.:
Applications of Functional Analysis in Mathematical Physics, AMS, 1963
* Lebedev, L.P. and Vorovich, I.I.:
Functional Analysis in Mechanics, Springer-Verlag, 2002
* Kolmogorov, A.N and Fomin, S.V.:
Elements of the Theory of Functions and Functional Analysis, Dover Publications, 1999
* Riesz, F. and Sz.-Nagy, B.:
Functional Analysis, Dover Publications, 1990
* Lax, P.:
Functional Analysis, Wiley-Interscience, 2002
* Rudin, W.:
Functional Analysis, McGraw-Hill Science, 1991
