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 (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:
(1) |
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
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; Hölder continuous functions
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; Lipschitz continuous functions
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Restrictions on integrability
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; integrable functions
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; square integrable functions
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functions (http://planetmath.org/LpSpace)
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Integrability of derivatives
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; functions of bounded variation, i.e. functions whose derivative is a measure
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; Sobolev space of -integrable functions which have -integrable derivatives of -th order. Space is a Hilbert space and is usually denoted by or .
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; functions with bounded mean oscillation. functions with vanishing mean oscillation
Restriction on growth
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Functions with polynomial growth
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; rapidly decreasing functions (Schwartz space)
Test function spaces
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; rapidly decreasing functions (Schwartz space)
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; smooth functions with compact support
Distribution spaces
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; distributions with compact support
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Piecewise properties
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; piecewise continuous functions
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; piecewise k times continuous differentiable functions
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; piecewise smooth functions (http://planetmath.org/PiecewiseSmooth)
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piecewise linear functions
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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):
Selected links
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The entry \htmladdnormallinkFunction spacehttp://en.wikipedia.org/wiki/Function_space at the \htmladdnormallinkWikipediahttp://en.wikipedia.org/.
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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/.
Title | function space |
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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 |