Editing Functional analysis (section)

==Normed vector spaces==
The basic and historically first class of spaces studied in functional analysis are [[complete space|complete]] [[normed vector space]]s over the [[real number|real]] or [[complex number]]s. Such spaces are called [[Banach space]]s. 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]], [[Reproducing kernel Hilbert space|machine learning]], [[partial differential equations]], and [[Fourier analysis]].

More generally, functional analysis includes the study of [[Fréchet space]]s and other [[topological vector space]]s not endowed with a norm.

An important object of study in functional analysis are the [[continuous function (topology)|continuous]] [[linear transformation|linear operators]] defined on Banach and Hilbert spaces. These lead naturally to the definition of [[C*-algebra]]s and other [[operator algebra]]s.

===Hilbert spaces===
[[Hilbert space]]s can be completely classified: there is a unique Hilbert space [[up to]] [[isomorphism]] for every [[cardinal number|cardinality]] of the [[orthonormal basis]].<ref>{{Cite book| last=Riesz|first=Frigyes| url=https://www.worldcat.org/oclc/21228994|title=Functional analysis|date=1990|publisher=Dover Publications| others = Béla Szőkefalvi-Nagy, Leo F. Boron|isbn=0-486-66289-6|edition=Dover |location=New York|oclc=21228994| pages = 195–199}}</ref> Finite-dimensional Hilbert spaces are fully understood in [[linear algebra]], and infinite-dimensional [[Separable space|separable]] Hilbert spaces are isomorphic to [[Sequence space#ℓp spaces|<math>\ell^{\,2}(\aleph_0)\,</math>]]. 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 space]]s 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 [[Lp space|<math>L^p</math>-spaces]] for any real number {{nowrap|<math>p\geq1</math>.}} Given also a measure <math>\mu</math> on set {{nowrap|<math>X</math>,}} then {{nowrap|<math>L^p(X)</math>,}} sometimes also denoted <math>L^p(X,\mu)</math> or {{nowrap|<math>L^p(\mu)</math>,}} has as its vectors equivalence classes <math>[\,f\,]</math> of [[Lebesgue-measurable function|measurable function]]s whose [[absolute value]]'s <math>p</math>-th power has finite integral; that is, functions <math>f</math> for which one has
<math display="block">\int_{X}\left|f(x)\right|^p\,d\mu(x) < \infty.</math>

If <math>\mu</math> is the [[counting measure]], then the integral may be replaced by a sum. That is, we require
<math display="block">\sum_{x\in X}\left|f(x)\right|^p < \infty .</math>

Then it is not necessary to deal with equivalence classes, and the space is denoted {{nowrap|<math>\ell^p(X)</math>,}} written more simply <math>\ell^p</math> in the case when <math>X</math> is the set of non-negative [[integer]]s.

In Banach spaces, a large part of the study involves the [[Continuous dual|dual space]]: the space of all [[continuous function (topology)|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.