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 , which contains functions acting from set to set , is denoted by . Evidently, . In the case when one usually writes only .
If the codomain is a vector space over field , then it is easy to define operations of the vector space on functions acting to in the following way:
where is an element of the field , and is an element of the domain (http://planetmath.org/Function) 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, 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
; functions with bounded mean oscillation. functions with vanishing mean oscillation
Restriction on growth
Test function spaces
; rapidly decreasing functions (Schwartz space)
; smooth functions with compact support
; piecewise continuous functions
; piecewise smooth functions (http://planetmath.org/PiecewiseSmooth)
piecewise linear functions
It is possible to attach a number which we call regularity index, to many of these spaces. If a space has a regularity index which is strictly less than the regularity index of , then (under some hypothesis on the domain of the functions) contains .
Here is a list of regularity indices ( is the dimension of the domain):
The entry \htmladdnormallinkFunction spacehttp://en.wikipedia.org/wiki/Function_space at the \htmladdnormallinkWikipediahttp://en.wikipedia.org/.
Chapter \htmladdnormallinkFunction spaceshttp://www.math.uiowa.edu/ dstewart/classes/22m176/dfs-notes/node2.html from the \htmladdnormallinkNotes on distributions and function spaceshttp://www.math.uiowa.edu/ dstewart/classes/22m176/dfs-notes/ by \htmladdnormallinkD. Stewarthttp://www.math.uiowa.edu/ dstewart/.
|Date of creation||2013-03-22 14:08:31|
|Last modified on||2013-03-22 14:08:31|
|Last modified by||matte (1858)|
|Synonym||space of functions|