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function space (Topic)

Generally speaking, a function space is a collection of functions satisfying some 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:

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

Restrictions on integrability

Integrability of derivatives

Restriction on growth

Test function spaces

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

Distribution spaces

Piecewise properties

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'$ $ -\infty$
$ \mathscr M$ $ -n$

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"function space" is owned by matte. [ full author list (9) ]
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Other names:  space of functions
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Cross-references: Wikipedia, indices, domain, hypothesis, strictly, index, regularity, number, simple functions, piecewise, Radon measures, support, compact, distributions, Schwartz space, decreasing, polynomial growth, bounded functions, mean, bounded, Hilbert space, order, Sobolev space, measure, derivative, functions of bounded variation, locally integrable function, essentially bounded functions, square, measurable functions, smooth functions with compact support, holomorphic functions, analytic functions, Lipschitz continuous, continuously differentiable, continuous functions, PlanetMath, topology, closed under, operations, field, vector space, contains, codomain, properties, functions, collection
There are 27 references to this entry.

This is version 34 of function space, born on 2004-02-10, modified 2008-02-05.
Object id is 5556, canonical name is FunctionSpaces.
Accessed 9938 times total.

Classification:
AMS MSC: 46-00 (Functional analysis :: General reference works )
 26-00 (Real functions :: General reference works )
 30H05 (Functions of a complex variable :: Spaces and algebras of analytic functions)
 54C35 (General topology :: Maps and general types of spaces defined by maps :: Function spaces)
Pending Errata and Addenda
None.
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Discussion
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forum policy
suggestion to "function spaces" by mathforever on 2005-04-28 11:38:26
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.
-------------------------------
knowledge can become a science
only with a help of mathematics
[ reply | up ]
HELP with linking policy by mathforever on 2005-01-22 11:29:16
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.
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