# function space

Generally speaking, a 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 integrability

• $L^{0}$; measurable functions

• $L^{1}$; integrable functions

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

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

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

• $L^{1}_{\scriptsize{\mbox{loc}}}(U)$; locally integrable function

## Integrability of derivatives

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

• $W^{m,p}(\Omega)$; 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 $H^{m}(\Omega)$.

• $BMO$; functions with bounded mean oscillation. $VMO$ functions with vanishing mean oscillation

## Test function spaces

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

• $\mathscr{D}$; smooth functions with compact support

## Distribution spaces

• $\mathscr{S}^{\prime}$; tempered distributions

• $\mathscr{D}^{\prime}$; distributions

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

• $\mathscr{M}$; Radon measures

## Piecewise properties

• $PC$; piecewise continuous functions

• $PC^{k}$; piecewise k times continuous differentiable functions

• $PC^{\infty}$; piecewise smooth functions (http://planetmath.org/PiecewiseSmooth)

• piecewise linear 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):

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