Editing Functional analysis (section)

===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.