Spaces of test functions and distributions

In <a href="/facts/Mathematical_analysis/XXeR26Ih">mathematical analysis</a>, the spaces of test functions and distributions are <a href="/facts/Topological_vector_space/XRex1ChN">topological vector spaces</a> (TVSs) that are used in the definition and application of <a href="/facts/Distribution_(mathematics)/yfaJ4mUp">distributions</a>. 
Test functions are usually <a href="/facts/Smooth_function/mffL4ch2">infinitely differentiable</a> <a href="/facts/Complex_number/2w7mqI7e">complex</a>-valued (or sometimes <a href="/facts/Real_number/R02gw5Pb">real</a>-valued) functions on a non-empty <a href="/facts/Open_set/QfTCIqMu">open subset</a> 
 
 
 
 U
 ⊆
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle U\subseteq \mathbb {R} ^{n}}
 
 that have <a href="/facts/Compact_space/d0cgXJH7">compact</a> <a href="/facts/Support_(mathematics)/BTuMGbsb">support</a>. 
The space of all test functions, denoted by 
 
 
 
 
 C
 
 c
 
 
 ∞
 
 
 (
 U
 )
 ,
 
 
 {\displaystyle C_{c}^{\infty }(U),}
 
 is endowed with a certain topology, called the canonical <a href="/facts/LF-space/voQHEeGI">LF-topology</a>, that makes 
 
 
 
 
 C
 
 c
 
 
 ∞
 
 
 (
 U
 )
 
 
 {\displaystyle C_{c}^{\infty }(U)}
 
 into a <a href="/facts/Complete_topological_vector_space/e6qtS76j">complete</a> <a href="/facts/Hausdorff_space/gbKIrusl">Hausdorff</a> <a href="/facts/Locally_convex_topological_vector_space/AGrPrayC">locally convex</a> <a href="/facts/Topological_vector_space/XRex1ChN">TVS</a>. 
The <a href="/facts/Strong_dual_space/4RI1SYl6">strong dual space</a> of 
 
 
 
 
 C
 
 c
 
 
 ∞
 
 
 (
 U
 )
 
 
 {\displaystyle C_{c}^{\infty }(U)}
 
 is called the space of distributions on 
 
 
 
 U
 
 
 {\displaystyle U}
 
 and is denoted by 
 
 
 
 
 
 
 D
 
 
 
 ′
 
 
 (
 U
 )
 :=
 
 
 (
 
 
 C
 
 c
 
 
 ∞
 
 
 (
 U
 )
 
 )
 
 
 b
 
 
 ′
 
 
 ,
 
 
 {\displaystyle {\mathcal {D}}^{\prime }(U):=\left(C_{c}^{\infty }(U)\right)_{b}^{\prime },}
 
 where the "
 
 
 
 b
 
 
 {\displaystyle b}
 
" subscript indicates that the <a href="/facts/Continuous_dual_space/4Uw4Knz1">continuous dual space</a> of 
 
 
 
 
 C
 
 c
 
 
 ∞
 
 
 (
 U
 )
 ,
 
 
 {\displaystyle C_{c}^{\infty }(U),}
 
 denoted by 
 
 
 
 
 
 (
 
 
 C
 
 c
 
 
 ∞
 
 
 (
 U
 )
 
 )
 
 
 ′
 
 
 ,
 
 
 {\displaystyle \left(C_{c}^{\infty }(U)\right)^{\prime },}
 
 is endowed with the <a href="/facts/Strong_dual_topology/4RI1SYl6">strong dual topology</a>.
There are other possible choices for the space of test functions, which lead to other different <a href="/facts/Vector_space/M1pxMLx2">spaces</a> of distributions. If 
 
 
 
 U
 =
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle U=\mathbb {R} ^{n}}
 
 then the use of <a href="/facts/Schwartz_space/B035faw7">Schwartz functions</a> as test functions gives rise to a certain subspace of 
 
 
 
 
 
 
 D
 
 
 
 ′
 
 
 (
 U
 )
 
 
 {\displaystyle {\mathcal {D}}^{\prime }(U)}
 
 whose elements are called tempered distributions. These are important because they allow the <a href="/facts/Fourier_transform/XXhKJtc8">Fourier transform</a> to be extended from "standard functions" to tempered distributions. The set of tempered distributions forms a <a href="/facts/Vector_space/M1pxMLx2">vector subspace</a> of the space of distributions 
 
 
 
 
 
 
 D
 
 
 
 ′
 
 
 (
 U
 )
 
 
 {\displaystyle {\mathcal {D}}^{\prime }(U)}
 
 and is thus one example of a space of distributions; there are many other spaces of distributions.
There also exist other major classes of test functions that are not subsets of 
 
 
 
 
 C
 
 c
 
 
 ∞
 
 
 (
 U
 )
 ,
 
 
 {\displaystyle C_{c}^{\infty }(U),}
 
 such as spaces of <a href="/facts/Analytic_function/JhL3z8FP">analytic test functions</a>, which produce very different classes of distributions. The theory of such distributions has a different character from the previous one because there are no analytic functions with non-empty compact support. Use of analytic test functions leads to <a href="/facts/Mikio_Sato/LUkFh98o">Sato</a>'s theory of <a href="/facts/Hyperfunction/MMFNjLhK">hyperfunctions</a>.

Spaces of test functions and distributions open-in-new

Spaces of test functions and distributions