Editing Distribution (mathematics) (section)

====Test functions as distributions====

The test functions are themselves locally integrable, and so define distributions. The space of test functions <math>C_c^\infty(U)</math> is sequentially [[dense (topology)|dense]] in <math>\mathcal{D}'(U)</math> with respect to the strong topology on <math>\mathcal{D}'(U).</math>{{sfn|Trèves|2006|pp=300-304}} This means that for any <math>T \in \mathcal{D}'(U),</math> there is a sequence of test functions, <math>(\phi_i)_{i=1}^\infty,</math> that converges to <math>T \in \mathcal{D}'(U)</math> (in its strong dual topology) when considered as a sequence of distributions. Or equivalently,
<math display=block>\langle \phi_i, \psi \rangle \to \langle T, \psi \rangle \qquad \text{ for all } \psi \in \mathcal{D}(U).</math>