Functional analysis

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.

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 measurable functions whose absolute value'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 dual space: the space of all continuous linear maps from the space into its underlying field, so-called functionals. A Banach space can be canonically identified with a subspace of its bidual, which is the dual of its dual space. The corresponding map is an isometry but in general not onto. A general Banach space and its bidual need not even be isometrically isomorphic in any way, contrary to the finite-dimensional situation. 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.

The uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerstones of the field. In its basic form, it asserts that for a family of continuous linear operators (and thus bounded operators) whose domain is a Banach space, pointwise boundedness is equivalent to uniform boundedness in operator norm.

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

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

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

Spectral theorem.[6] 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 operator theory; see also the spectral measure.

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.

The Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed vector space to make the study of the dual space "interesting".

Hahn–Banach theorem:[7] 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

${\displaystyle \psi (x)=\varphi (x)\qquad \forall x\in U,}$

${\displaystyle \psi (x)\leq p(x)\qquad \forall x\in V.}$

The open mapping theorem, also known as the Banach–Schauder theorem (named after Stefan Banach and Juliusz Schauder), is a fundamental result which states that if a continuous linear operator between Banach spaces is surjective then it is an open map. More precisely,:[7]

Open mapping theorem. If ${\displaystyle X}$ and ${\displaystyle Y}$ are Banach spaces and ${\displaystyle A:X\to Y}$ is a surjective continuous linear operator, then ${\displaystyle A}$ is an open map (that is, if ${\displaystyle U}$ is an open set in ${\displaystyle X}$, then ${\displaystyle A(U)}$ is open in ${\displaystyle Y}$).

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 normed space, but is true if ${\displaystyle X}$ and ${\displaystyle Y}$ are taken to be Fréchet spaces.

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 continuous.[8]