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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 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 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, 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 $H^{{m}}(\Omega)$.
Restriction on growth

$B$; bounded functions

Functions with polynomial growth

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

$\mathscr{M}$; Radon measures
Piecewise properties

$PC$; piecewise continuous functions

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

$PC^{{\infty}}$; piecewise smooth functions

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):
$C$  $0$ 

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

The entry Function space at the Wikipedia.

Chapter Function spaces from the Notes on distributions and function spaces by D. Stewart.
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Comments
HELP with linking policy
Hello everybody!
I tried to understand what to do when there is a concept which is linked wrongly, but I failed to apply it to the situation in this entry (I also didn't understand the whole policy) :(. Thus let me ask to help with it. So here is the problem:
in the entry 'function spaces' there is a word 'domain' which is meant domain of the function. There are 4 'domain'related entries on PM, and the linker links this word not to domain of the function. I tried in the linking policy to type:
priority 10 domain
priority 200 domain
but nothing changed. So what should be done here?
Thanks in advance.
Re: HELP with linking policy
A surefire way of making sure that the links point to the right entries is to use the \PMlinkname or \PMlinkid commands. For instance, if I want to be sure that the word "binomial" linked to the entry "binomial coefficient", I would tpye \PMlinkname{binomial}{BinomialCoefficient} (the canonical name of the entry coes in the second slot) or \PMlinkid{binomial}{273} (the canonical number of the entry "binomial coefficient is 273), whilst if I wanted it to link to the entry "binomial theorem", I would type \PMlinkname{binomial}{BinomialTheorem} or \PMlinkid{binomial}{247}.
Re: HELP with linking policy
Your main problem is that linking priority is an element of the metadata of the entry that *defines* the term in question (in this case, the entry that defines "domain" in the function sense). So, you need to file a correction to the owner of that domain entry asking them to add a linking policy entry. This policy would basically look like:
priority 10
(the term isn't needed since its the title).
I'm not sure what you were trying to achieve with the "priority 200" line, though.
I think, to do what you want to do (create one default/primary sense of "domain"), you only need the single linking policy statement in that one entry.
Hope this helps.
apk
Re: HELP with linking policy
Thanks for reply. As it is written in the documentation one should avoid this way of 'brutal' ;) forcing the linking. Since then in EVERY entry with such problem one needs to do use the commands \PMlinkname,\PMlinkid (but thanks for your examples, now it is clear for me how to use them). Instead there is this concept of linking policy, where one changes something only in the linked entry, and then everything should work auomatically. However the mechanism of the linking policy is unclear for me :(
still questions (Re: HELP with linking policy)
Thanks for reply!
> you need to file a correction to
> the owner of that domain entry
> asking them to add a linking policy entry
I'll do this. Now the 'linking policy' thing is a little bit more clear to me, but still not complete: it just seems too easy to work :). Assume the owner will change priority to higher. How then linker will decide to link the 'domain' here to the 'domain' in function entry? Will it look on the classification? Then he should decide that classification of the 'function spaces' which is like
"functional analysis", "real functions", "functions of a complex variable"
is more related to
"math.logic and foundations"
which is classification of the domain function, then to
"functions of a complex variable"
which is classification of the 'wrong' domain link. If the linker would ask me, then I would say that the second option is more preferable, since these two entries have the same classifications and thus wrong link will remain. Thus it seems to me that linking policy won't work here, or?
Re: still questions (Re: HELP with linking policy)
The priority element of the linking policy only gets used if classification doesn't narrow things down. So if there's a category match with one of the "wrong" senses of domain, this won't help.
If this is the case, there is likely a categorization problem somewhere.
apk
Re: still questions (Re: HELP with linking policy)
> if there's a category match
> with one of the "wrong" senses of domain,
> [linking policy] won't help.
Yea... I see now. Then I'll use the method proposed by Ray: command \PMlinkname. Since the categorizations are right: categorization of the right domain entry is "math.logic" which is far away from categorization of function spaces.
Many thanks for help.
Serg.
Re: still questions (Re: HELP with linking policy)
Wait a minute =)
The categories dont *have* to match. Categorization is only a problem if there is a *wrong* match. So, if the categorization of function spaces matches one of the domain categorizations, then linking will pick that wrong sense of "domain", regardless of the priority of the right sense.
However, if there is no match, the priority should still help. Can you look at the entry with the wrong "domain" sense and confirm that it has a matching categorization? If not, setting the link priority on the correct "domain" entry should fix the problem.
Manual linking commands should really be avoided when not necessary.
apk
Re: still questions (Re: HELP with linking policy)
> Can you look at the entry with
> the wrong "domain" sense and
> confirm that it has a matching categorization?
yea, that's what I already did. 'Function spaces' have categorizations:
4600 (Functional analysis :: General reference works )
2600 (Real functions :: General reference works )
30H05 (Functions of a complex variable :: Spaces and algebras of analytic functions)
the right domain entry (domain of the function) has categorization:
03E20 (Mathematical logic and foundations :: Set theory :: Other classical set theory )
and the wrong domain entry (domain as a set) has categorization:
3000 (Functions of a complex variable :: General reference works )
So, you see that the wrong one has much more match with 'f.spaces' than the right one :(
May be one should add some more categorizations to the right domain entry... What will happen if there will be two domain entries with categorizations like
(domain as a function)
03E20 (Mathematical logic and foundations :: Set theory :: Other classical set theory )
4600 (Functional analysis :: General reference works )
2600 (Real functions :: General reference works )
3000 (Functions of a complex variable :: General reference works )
(domain as set)
3000 (Functions of a complex variable :: General reference works )
and some categorization from the set theory
So, then both have matching categorization with "f.spaces", what will be linked?
suggestion to "function spaces"
Let me make a suggestion to the entry "function spaces". Due to the recent changes in this entry, there appears such a notion as regularity index. So, my suggestion is to think of some more motivated and explained introduction of this thing, since for the moment its presentation is rather vague.
Regards
Serg.

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