Editing Distribution (mathematics) (section)

===Distributions with compact support===

The inclusion map <math>\operatorname{In}: C_c^\infty(U) \to C^\infty(U)</math> is a continuous injection whose image is dense in its codomain, so the [[Transpose of a linear map|transpose map]] <math>{}^{t}\operatorname{In}: (C^\infty(U))'_b \to \mathcal{D}'(U) = (C_c^\infty(U))'_b</math> is also a continuous injection. Thus the image of the transpose, denoted by <math>\mathcal{E}'(U),</math> forms a space of distributions.{{sfn|Trèves|2006|pp=255-257}}

The elements of <math>\mathcal{E}'(U) = (C^\infty(U))'_b</math> can be identified as the space of distributions with compact support.{{sfn|Trèves|2006|pp=255-257}} Explicitly, if <math>T</math> is a distribution on {{mvar|U}} then the following are equivalent,

* <math>T \in \mathcal{E}'(U).</math>
* The support of <math>T</math> is compact.
* The restriction of <math>T</math> to <math>C_c^\infty(U),</math> when that space is equipped with the subspace topology inherited from <math>C^\infty(U)</math> (a coarser topology than the canonical LF topology), is continuous.{{sfn|Trèves|2006|pp=255-257}}
* There is a compact subset {{mvar|K}} of {{mvar|U}} such that for every test function <math>\phi</math> whose support is completely outside of {{mvar|K}}, we have <math>T(\phi) = 0.</math>

Compactly supported distributions define continuous linear functionals on the space <math>C^\infty(U)</math>; recall that the topology on <math>C^\infty(U)</math> is defined such that a sequence of test functions <math>\phi_k</math> converges to 0 if and only if all derivatives of <math>\phi_k</math> converge uniformly to 0 on every compact subset of {{mvar|U}}. Conversely, it can be shown that every continuous linear functional on this space defines a distribution of compact support. Thus compactly supported distributions can be identified with those distributions that can be extended from <math>C_c^\infty(U)</math> to <math>C^\infty(U).</math>