function space (original) (raw)

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 ℱ, which contains functions acting from set X to set Y, is denoted by ℱ⁢(X,Y). Evidently, ℱ⁢(X,Y)⊆YX. In the case when Y=ℝ one usually writes only ℱ⁢(X).

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

where α is an element of the field K, and x 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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  Ck,α; Hölder continuous functions
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Restrictions on integrability

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  L1; integrable functions
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  Lp functions (http://planetmath.org/LpSpace)
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Integrability of derivatives

Restriction on growth

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  Functions with polynomial growth
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Test function spaces

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  𝒮; rapidly decreasing functions (Schwartz space)
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  𝒟; smooth functions with compact support

Distribution spaces

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  ℰ′; distributions with compact support
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Piecewise properties

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  P⁢C∞; 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 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.

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/.