Functional analysis

The usage of the word functional 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.^[1][2] 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.^[3][4] 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 theories 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

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

Hilbert spaces can be completely classified: there is a unique Hilbert space up to isomorphism for every cardinality of the orthonormal basis.^[5] Finite-dimensional Hilbert spaces are fully understood in linear algebra, and infinite-dimensional separable Hilbert spaces are isomorphic to ${\displaystyle \ell ^{\,2}(\aleph _{0})\,}$. 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 ${\displaystyle L^{p}}$-spaces for any real number ${\displaystyle p\geq 1}$. Given also a measure ${\displaystyle \mu }$ on set ${\displaystyle X}$, then ${\displaystyle L^{p}(X)}$, sometimes also denoted ${\displaystyle L^{p}(X,\mu )}$ or ${\displaystyle L^{p}(\mu )}$, has as its vectors equivalence classes ${\displaystyle [\,f\,]}$ of

's ${\displaystyle p}$-th power has finite integral; that is, functions ${\displaystyle f}$ for which one has

$${\displaystyle \int _{X}\left|f(x)\right|^{p}\,d\mu (x)<\infty .}$$

If ${\displaystyle \mu }$ is the counting measure, then the integral may be replaced by a sum. That is, we require

$${\displaystyle \sum _{x\in X}\left|f(x)\right|^{p}<\infty .}$$

Then it is not necessary to deal with equivalence classes, and the space is denoted ${\displaystyle \ell ^{p}(X)}$, written more simply ${\displaystyle \ell ^{p}}$ in the case when ${\displaystyle X}$ is the set of non-negative integers.

In Banach spaces, a large part of the study involves the

Also, the notion of derivative can be extended to arbitrary functions between Banach spaces. See, for instance, the Fréchet derivative article.

Linear functional analysis

^[6]

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Major and foundational results

There are four major theorems which are sometimes called the four pillars of functional analysis:

• the Hahn–Banach theorem
• the open mapping theorem
• the closed graph theorem
• the
  Banach–Steinhaus theorem
  .

Important results of functional analysis include:

Uniform boundedness principle

The

The theorem was first published in 1927 by Stefan Banach and Hugo Steinhaus but it was also proven independently by Hans Hahn.

Theorem (Uniform Boundedness Principle) — Let ${\displaystyle X}$ be a Banach space and ${\displaystyle Y}$ be a normed vector space. Suppose that ${\displaystyle F}$ is a collection of continuous linear operators from ${\displaystyle X}$ to ${\displaystyle Y}$. If for all ${\displaystyle x}$ in ${\displaystyle X}$ one has

$${\displaystyle \sup \nolimits _{T\in F}\|T(x)\|_{Y}<\infty ,}$$

then

$${\displaystyle \sup \nolimits _{T\in F}\|T\|_{B(X,Y)}<\infty .}$$

Spectral theorem

There are many theorems known as the spectral theorem, but one in particular has many applications in functional analysis.

Spectral theorem^[7] — Let ${\displaystyle A}$ be a bounded self-adjoint operator on a Hilbert space ${\displaystyle H}$. Then there is a measure space ${\displaystyle (X,\Sigma ,\mu )}$ and a real-valued

essentially bounded

measurable function ${\displaystyle f}$ on ${\displaystyle X}$ and a unitary operator ${\displaystyle U:H\to L_{\mu }^{2}(X)}$ such that

$${\displaystyle U^{*}TU=A}$$

where T is the multiplication operator:

$${\displaystyle [T\varphi ](x)=f(x)\varphi (x).}$$

and ${\displaystyle \|T\|=\|f\|_{\infty }}$.

This is the beginning of the vast research area of functional analysis called

There is also an analogous spectral theorem for bounded normal operators on Hilbert spaces. The only difference in the conclusion is that now ${\displaystyle f}$ may be complex-valued.

Hahn–Banach theorem

The

Hahn–Banach theorem:[8] — If ${\displaystyle p:V\to \mathbb {R} }$ is a sublinear function, and ${\displaystyle \varphi :U\to \mathbb {R} }$ is a

linear functional on a linear subspace

${\displaystyle U\subseteq V}$ which is dominated by ${\displaystyle p}$ on ${\displaystyle U}$; that is,

$${\displaystyle \varphi (x)\leq p(x)\qquad \forall x\in U}$$

then there exists a linear extension ${\displaystyle \psi :V\to \mathbb {R} }$ of ${\displaystyle \varphi }$ to the whole space ${\displaystyle V}$ which is dominated by ${\displaystyle p}$ on ${\displaystyle V}$; that is, there exists a linear functional ${\displaystyle \psi }$ such that

$${\displaystyle {\begin{aligned}\psi (x)&=\varphi (x)&\forall x\in U,\\\psi (x)&\leq p(x)&\forall x\in V.\end{aligned}}}$$

The

The proof uses the Baire category theorem, and completeness of both ${\displaystyle X}$ and ${\displaystyle Y}$ is essential to the theorem. The statement of the theorem is no longer true if either space is just assumed to be a

${\displaystyle X}$

${\displaystyle Y}$

Closed graph theorem

The closed graph theorem states the following: If ${\displaystyle X}$ is a topological space and ${\displaystyle Y}$ is a compact Hausdorff space, then the graph of a linear map ${\displaystyle T}$ from ${\displaystyle X}$ to ${\displaystyle Y}$ is closed if and only if ${\displaystyle T}$ is

Foundations of mathematics considerations

Most spaces considered in functional analysis have infinite dimension. To show the existence of a

Functional analysis in its present form[update] includes the following tendencies:

• Abstract analysis. An approach to analysis based on topological groups, topological rings, and topological vector spaces.
• Geometry of
  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, for example, Israel Gelfand, to include most types of representation theory.