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 $ℱ$, which contains functions acting from set $X$ to set $Y$, is denoted by $ℱ(X,Y)$. Evidently, $ℱ(X,Y)\subseteq Y^X$. In the case when $Y=ℝ$ one usually writes only $ℱ(X)$.

If the codomain $Y$ is a vector space over field $K$, then it is easy to defineoperations of the vector space on functions acting to $Y$ in the following way:

(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 thusare 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 listof function spaces that already have beendefined on PlanetMath (or should be), a gallery of functionspaces if you like.

Restrictions on smoothness

• $C$; continuous functions

• $C^k$; $k$ times continuously differentiable functions

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

• $\mathrm{Lip}$; Lipschitz continuous functions

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

• $C^{\omega }$; analytic functions

• $𝒪(G)$; holomorphic functions

• $C_c^{\mathrm{\infty }}$ or $𝒟$; smooth functions with compact support

Restrictions on integrability

• $L^0$; measurable functions

• $L^1$; integrable functions

• $L^2$; square integrable functions

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

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

• $L_{\text{loc}}^1(U)$; locally integrable function

Integrability of derivatives

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

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

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

Restriction on growth

• $B$; bounded functions

• Functions with polynomial growth

• $𝒮$; rapidly decreasing functions (Schwartz space)

Test function spaces

• $𝒮$; rapidly decreasing functions (Schwartz space)

• $𝒟$; smooth functions with compact support

Distribution spaces

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

• ${𝒟}^{\prime }$; distributions

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

• $ℳ$; Radon measures

Piecewise properties

• $PC$; piecewise continuous functions

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

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

• piecewise linear functions

• simple functions

It is possible to attach a number which we callregularity 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):

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

Selected links

• The entry Function spacehttp://en.wikipedia.org/wiki/Function_spaceat the Wikipediahttp://en.wikipedia.org/.

• Chapter Function spaceshttp://www.math.uiowa.edu/ dstewart/classes/22m176/dfs-notes/node2.htmlfrom theNotes on distributions and function spaceshttp://www.math.uiowa.edu/ dstewart/classes/22m176/dfs-notes/byD. Stewarthttp://www.math.uiowa.edu/ dstewart/.