Editing Function space (section)

==Functional analysis==
[[Functional analysis]] is organized around adequate techniques to bring function spaces as [[topological vector space]]s within reach of the ideas that would apply to [[normed space]]s of finite dimension. Here we use the real line as an example domain, but the spaces below exist on suitable open subsets <math>\Omega \subseteq \R^n</math>

*<math>C(\R)</math> [[continuous functions]] endowed with the [[uniform norm]] topology
*<math>C_c(\R)</math> continuous functions with [[Support (mathematics)#Compact support|compact support]]
* <math>B(\R)</math> [[bounded function]]s
* <math>C_0(\R)</math> continuous functions which vanish at infinity
* <math>C^r(\R)</math> continuous functions that have ''r'' continuous derivatives.
* <math>C^{\infty}(\R)</math> [[smooth functions]]
* <math>C^{\infty}_c(\R)</math> [[smooth functions]] with [[Support (mathematics)#Compact support|compact support]] (i.e. the set of [[bump function]]s)
*<math>C^\omega(\R)</math> [[Analytic function|real analytic functions]]
*<math>L^p(\R)</math>, for <math>1\leq p \leq \infty</math>, is the [[Lp space|L<sup>p</sup> space]] of [[Measurable function|measurable]] functions whose ''p''-norm <math display="inline">\|f\|_p = \left( \int_\R |f|^p \right)^{1/p}</math> is finite
*<math>\mathcal{S}(\R)</math>, the [[Schwartz space]] of [[rapidly decreasing]] [[smooth functions]] and its continuous dual, <math>\mathcal{S}'(\R)</math> [[tempered distributions]]
*<math>D(\R)</math> compact support in limit topology
* <math>W^{k,p}</math> [[Sobolev space]] of functions whose [[Weak_derivative|weak derivatives]] up to order ''k'' are in <math>L^p</math>
* <math>\mathcal{O}_U</math> holomorphic functions
* linear functions
* piecewise linear functions
* continuous functions, compact open topology
* all functions, space of pointwise convergence 
* [[Hardy space]]
* [[Hölder space]]
* [[Càdlàg]] functions, also known as the [[Anatoliy Skorokhod|Skorokhod]] space
* <math>\text{Lip}_0(\R)</math>, the space of all [[Lipschitz continuous|Lipschitz]] functions on <math>\R</math> that vanish at zero.