functional analysis

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.

0.1

Normed vector spaces

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 FrÃÂ©chet 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

(

http://planetmath.org/CAlgebra

) and other operator algebras.

0.2

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 the

invariant subspace problem

, which conjectures that every

operator

on a Hilbert space has a non-trivial

invariant subspace

. Many special cases have already been proven.

0.3

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

L

p

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 FrÃÂ©chet derivative.

0.4

Major and foundational results

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.

0.5

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, which relies on the

axiom of choice

that is

strictly

weaker than the

Boolean prime ideal theorem

.

0.6

Points of view

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.

This entry was adapted from the Wikipedia article

http://en.wikipedia.org/wiki/Functional_analysis

Functional analysis as of December 18, 2006.

Title

functional analysis

Canonical

name

FunctionalAnalysis

Date of creation

2013-03-22 16:28:19

Last modified on

2013-03-22 16:28:19

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

12

Author

PrimeFan (13766)

Entry type

Topic

Classification

msc 26E35

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