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:

$\begin{array}[]{rcl}(\alpha\cdot f)\,(x)&=&\alpha\cdot f(x)\\ (f+g)\,(x)&=&f(x)+g(x)\end{array}$

(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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$C^{\infty}$ ; smooth functions
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$C^{\omega}$ ; analytic functions
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$\mathcal{O}(G)$ ; holomorphic functions
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$C_{c}^{\infty}$ 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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$L^{\infty}$ ; essentially bounded functions
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$L^{1}_{\scriptsize{\mbox{loc}}}(U)$ ; locally integrable function

Integrability of derivatives

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$B V$ ; functions of bounded variation, i.e. functions whose derivative is a measure
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$W^{m,p}(\Omega)$ ; Sobolev space of $p$ -integrable functions which have $p$ -integrable derivatives of $m$ -th order. Space $W^{m,2}(\Omega)$ is a Hilbert space and is usually denoted by $W^{m}(\Omega)$ or $H^{m}(\Omega)$ .
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$B M O$ ; functions with bounded mean oscillation. $V M O$ 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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$\mathscr{S}$ ; rapidly decreasing functions (Schwartz space)

Test function spaces

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$\mathscr{S}$ ; rapidly decreasing functions (Schwartz space)
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$\mathscr{D}$ ; smooth functions with compact support

Distribution spaces

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$\mathscr{S}^{\prime}$ ; tempered distributions
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$\mathscr{D}^{\prime}$ ; distributions
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$\mathscr{E}^{\prime}$ ; distributions with compact support
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$\mathscr{M}$ ; Radon measures

Piecewise properties

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$P C$ ; piecewise continuous functions
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$PC^{k}$ ; piecewise k times continuous differentiable functions
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$PC^{\infty}$ ; piecewise smooth functions (http://planetmath.org/PiecewiseSmooth)
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piecewise linear functions
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simple functions

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):

$C$	$0$
$C^{k}$	$k$
$C^{\infty}$	$\infty$
$C^{\omega}$	$\infty$
$C^{k,\alpha}$	$k+\alpha$
$\mathrm{Lip}$	$1$
$L^{p}$	$-n/p$
$L^{\infty}$	$0$
$W^{k,p}$	$k-n/p$
$W^{k,\infty}$	$k$
$B V$	$0$
$\mathscr{D}^{\prime}$	$-\infty$
$\mathscr{M}$	$-n$

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/dfs-notes/node2.html from the \htmladdnormallinkNotes on distributions and function spaceshttp://www.math.uiowa.edu/ dstewart/classes/22m176/dfs-notes/ by \htmladdnormallinkD. Stewarthttp://www.math.uiowa.edu/ dstewart/.

Title	function space
Canonical name	FunctionSpace
Date of creation	2013-03-22 14:08:31
Last modified on	2013-03-22 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 26-00
Classification	msc 46-00
Classification	msc 30H05
Synonym	space of functions