function space

Generally speaking, a function spaceMathworldPlanetmath is a collectionMathworldPlanetmath of functionsMathworldPlanetmath 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 , which contains functions acting from set X to set Y, is denoted by (X,Y). Evidently, (X,Y)YX. In the case when Y= one usually writes only (X).

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

(αf)(x)=αf(x)(f+g)(x)=f(x)+g(x) (1)

where α 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 topologyMathworldPlanetmath.

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 smoothness

Restrictions on integrability

Integrability of derivatives

Restriction on growth

Test function spaces

  • 𝒮; rapidly decreasing functions (Schwartz space)

  • 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 hypothesisMathworldPlanetmath on the domain of the functions) X contains Y.

Here is a list of regularity indices (n is the dimensionMathworldPlanetmathPlanetmath of the domain):

C 0
Ck k
Ck,α k+α
Lip 1
Lp -n/p
L 0
Wk,p k-n/p
Wk, k
BV 0
𝒟 -

Selected links

  • The entry \htmladdnormallinkFunction space at the \htmladdnormallinkWikipedia

  • Chapter \htmladdnormallinkFunction spaces dstewart/classes/22m176/dfs-notes/node2.html from the \htmladdnormallinkNotes on distributions and function spaces dstewart/classes/22m176/dfs-notes/ by \htmladdnormallinkD. Stewart dstewart/.

Title function space
Canonical name FunctionSpace
Date of creation 2013-03-22 14:08:31
Last modified on 2013-03-22 14:08:31
Owner matte (1858)
Last modified by matte (1858)
Numerical id 38
Author matte (1858)
Entry type Topic
Classification msc 54C35
Classification msc 26-00
Classification msc 46-00
Classification msc 30H05
Synonym space of functions