function space
Generally speaking, a function space^{} is a collection^{} of functions^{} satisfying certain properties. Typically, these properties are topological in nature, and hence the word “space”. Usually, functions in a function space have a common domain (http://planetmath.org/Function) and codomain. Thus, a function space $\mathcal{F}$, which contains functions acting from set $X$ to set $Y$, is denoted by $\mathcal{F}(X,Y)$. Evidently, $\mathcal{F}(X,Y)\subseteq {Y}^{X}$. In the case when $Y=\mathbb{R}$ one usually writes only $\mathcal{F}(X)$.
If the codomain $Y$ is a vector space over field $K$, then it is easy to define operations^{} of the vector space on functions acting to $Y$ in the following way:
$$\begin{array}{ccc}\hfill (\alpha \cdot f)(x)& \hfill =\hfill & \alpha \cdot f(x)\hfill \\ \hfill (f+g)(x)& \hfill =\hfill & f(x)+g(x)\hfill \end{array}$$  (1) 
where $\alpha $ is an element of the field $K$, and $x$ is an element of the domain (http://planetmath.org/Function) of functions. One usually consider function spaces which are closed under operations (1) and thus are vector spaces. Function spaces are also often equipped with some topology^{}.
Below is a list of function spaces, to entries where they are defined, and notation for these.
The main purpose of this entry is to give a list of function spaces that already have been defined on PlanetMath (or should be), a gallery of function spaces if you like.
Restrictions on smoothness

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$C$; continuous functions^{}

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${C}^{k}$; $k$ times continuously differentiable functions

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${C}^{k,\alpha}$; Hölder continuous functions

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$\mathrm{Lip}$; Lipschitz continuous functions

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${C}^{\mathrm{\infty}}$; smooth functions^{}

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${C}^{\omega}$; analytic functions^{}

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$\mathcal{O}(G)$; holomorphic functions^{}

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${C}_{c}^{\mathrm{\infty}}$ or $\mathcal{D}$; smooth functions with compact support
Restrictions on integrability

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${L}^{0}$; measurable functions^{}

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${L}^{1}$; integrable functions

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${L}^{2}$; square integrable functions

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${L}^{p}$ functions (http://planetmath.org/LpSpace)

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${L}^{\mathrm{\infty}}$; essentially bounded functions

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${L}_{\text{loc}}^{1}(U)$; locally integrable function
Integrability of derivatives

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$BV$; functions of bounded variation, i.e. functions whose derivative^{} is a measure^{}

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${W}^{m,p}(\mathrm{\Omega})$; Sobolev space^{} of $p$integrable functions which have $p$integrable derivatives of $m$th order. Space ${W}^{m,2}(\mathrm{\Omega})$ is a Hilbert space^{} and is usually denoted by ${W}^{m}(\mathrm{\Omega})$ or ${H}^{m}(\mathrm{\Omega})$.

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$BMO$; functions with bounded mean oscillation. $VMO$ functions with vanishing mean oscillation
Restriction on growth

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$B$; bounded functions

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Functions with polynomial growth

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$\mathcal{S}$; rapidly decreasing functions (Schwartz space^{})
Test function spaces

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$\mathcal{S}$; rapidly decreasing functions (Schwartz space)

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$\mathcal{D}$; smooth functions with compact support
Distribution spaces

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${\mathcal{S}}^{\prime}$; tempered distributions

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${\mathcal{D}}^{\prime}$; distributions^{}

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${\mathcal{E}}^{\prime}$; distributions with compact support

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$\mathcal{M}$; Radon measures^{}
Piecewise properties

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$PC$; piecewise continuous functions

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$P{C}^{k}$; piecewise k times continuous differentiable functions

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$P{C}^{\mathrm{\infty}}$; piecewise smooth functions (http://planetmath.org/PiecewiseSmooth)

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piecewise linear functions
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It is possible to attach a number which we call regularity index, to many of these spaces. If a space $X$ has a regularity index which is strictly less than the regularity index of $Y$, then (under some hypothesis^{} on the domain of the functions) $X$ contains $Y$.
Here is a list of regularity indices ($n$ is the dimension^{} of the domain):
$C$  $0$ 

${C}^{k}$  $k$ 
${C}^{\mathrm{\infty}}$  $\mathrm{\infty}$ 
${C}^{\omega}$  $\mathrm{\infty}$ 
${C}^{k,\alpha}$  $k+\alpha $ 
$\mathrm{Lip}$  $1$ 
${L}^{p}$  $n/p$ 
${L}^{\mathrm{\infty}}$  $0$ 
${W}^{k,p}$  $kn/p$ 
${W}^{k,\mathrm{\infty}}$  $k$ 
$BV$  $0$ 
${\mathcal{D}}^{\prime}$  $\mathrm{\infty}$ 
$\mathcal{M}$  $n$ 
Selected links

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The entry \htmladdnormallinkFunction spacehttp://en.wikipedia.org/wiki/Function_space at the \htmladdnormallinkWikipediahttp://en.wikipedia.org/.

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Chapter \htmladdnormallinkFunction spaceshttp://www.math.uiowa.edu/ dstewart/classes/22m176/dfsnotes/node2.html from the \htmladdnormallinkNotes on distributions and function spaceshttp://www.math.uiowa.edu/ dstewart/classes/22m176/dfsnotes/ by \htmladdnormallinkD. Stewarthttp://www.math.uiowa.edu/ dstewart/.
Title  function space 

Canonical name  FunctionSpace 
Date of creation  20130322 14:08:31 
Last modified on  20130322 14:08:31 
Owner  matte (1858) 
Last modified by  matte (1858) 
Numerical id  38 
Author  matte (1858) 
Entry type  Topic 
Classification  msc 54C35 
Classification  msc 2600 
Classification  msc 4600 
Classification  msc 30H05 
Synonym  space of functions 