Restriction of a distribution ($u{|}_V$)

For open sets $V\subseteq U$ and $u\in {\mathcal{D}}^{\prime }(U)$, define $⟨u{|}_V,\varphi ⟩=⟨u,\tilde{\varphi }⟩$, where $\tilde{\varphi }$ extends the test function $\varphi \in \mathcal{D}(V)$ 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.

Target: Locality of distributions. Parent: Distribution (mathematical analysis). Intended file: distribution-theory.bigb.