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 and codomain. Thus, a function space $\EuScript{F}$ , which contains functions acting from set $X$ to set $Y$ , is denoted by $\EuScript{F}(X,Y)$ . Evidently, $\EuScript{F}(X,Y)\subseteq Y^X$ . In the case when $Y=\mathbb{R}$ one usually writes only $\EuScript{F}(X)$ .

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:

(1)

Below is a list of function spaces, links 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

• $C$ ; continuous functions
• $C^k$ ; $k$ times continuously differentiable functions
• $C^{k,\alpha}$ ; Hölder continuous functions
• $\mathrm{Lip}$ ; Lipschitz continuous functions
• $C^\infty$ ; smooth functions
• $C^\omega$ ; analytic functions
• $\mathcal{O}(G)$ ; holomorphic functions
• $C_c^\infty$ or $\mathcal{D}$ ; smooth functions with compact support

Restrictions on integrability

• $L^0$ ; measurable functions
• $L^1$ ; integrable functions
• $L^2$ ; square integrable functions
• $L^p$ functions
• $L^\infty$ ; 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

Restriction on growth

• $B$ ; bounded functions
• Functions with polynomial growth
• ; rapidly decreasing functions (Schwartz space)

Test function spaces

• ; rapidly decreasing functions (Schwartz space)
• ; smooth functions with compact support

Distribution spaces

• ; tempered distributions
• ; distributions
• ; distributions with compact support
• ; Radon measures

Piecewise properties

• $PC$ ; piecewise continuous functions
• $PC^k$ ; piecewise k times continuous differentiable functions
• $PC^{\infty}$ ; piecewise smooth functions
• piecewise linear functions
• 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):

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

Selected links

• The entry Function space at the Wikipedia.
• Chapter Function spaces from the Notes on distributions and function spaces by D. Stewart.