= Restriction of a distribution
{title2=$u|_V$}
= Distributional restriction
{synonym}
For <open sets> $V\subseteq U$ and $u\in\mathcal D'(U)$, define $\langle u|_V,\phi\rangle=\langle u,\widetilde\phi\rangle$, where $\widetilde\phi$ extends the <test function> $\phi\in\mathcal D(V)$ by zero. Because its <compact support> lies inside $V$, this extension is smooth. Extension preserves convergence in the <space of test functions>, proving continuity of the restricted <distribution>. Successive restrictions agree with direct restriction.
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Target: Locality of distributions. Parent: Distribution (mathematical analysis). Intended file: distribution-theory.bigb.
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