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 (e.g.
inner product,
norm,
topology, etc.) and the
linear functions defined on 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
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
,
unitary
Unitary may refer to:
Mathematics
* Unitary divisor
* Unitary element
* Unitary group
* Unitary matrix
* Unitary morphism
* Unitary operator
* Unitary transformation
* Unitary representation In mathematics, a unitary representation of a grou ...
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
Functional may refer to:
* Movements in architecture:
** Functionalism (architecture)
** Form follows function
* Functional group, combination of atoms within molecules
* Medical conditions without currently visible organic basis:
** Functional sy ...
'' as a noun goes back to the
calculus of variations, implying a
function whose argument is a function. The term 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 on 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
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
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,
machine learning,
partial differential equations, and
Fourier analysis.
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
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
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 In mathematics, an invariant subspace of a linear mapping ''T'' : ''V'' → ''V '' i.e. from some vector space ''V'' to itself, is a subspace ''W'' of ''V'' that is preserved by ''T''; that is, ''T''(''W'') ⊆ ''W''.
General desc ...
. 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