Lemma 5

Let $\mathfrak{F}$ be a filter on $\mathscr{D}'(U)$ having a countable basis or containing a simply bounded set. Then $\mathfrak{F}$ converges to a distribution if and only if $\langle f, \varphi \rangle$ converges according to $\mathfrak{F}$ for every test function $\varphi$ on U.

In particular, a sequence $(f_m)_{m\in\mathbf{N}}$ of distributions converges to a distribution $f$ if and only if, for every $\varphi \in \mathscr{D}(U)$, the sequence $(\langle f_m, \varphi \rangle )_{m\in\mathbf{N}}$ converges to $\langle f, \varphi \rangle$.

Since $\mathscr{D}'(U)$ is a Montel space, hence barrelled, this follows from the Banach-Steinhaus theorem (EVT, III, p. 26, cor. 3 of th. 1).

Let V be an open set contained in U. The transpose of the canonical linear mapping of $\mathscr{D}(V)$ into $\mathscr{D}(U)$ is a continuous linear mapping of $\mathscr{D}'(U)$ into $\mathscr{D}'$(V), called the restriction of distributions from U to V and denoted by $r_{VU}$, or sometimes $r_{V,U}$.

One has $r_{VV}= 1_{\mathscr{D}'(V)}$. If $W\subset V\subset U$ are open sets in U, then one has $r_{WV}\circ r_{VU}=r_{WU}$. In other words, if $\mathscr{T}$ denotes the topology on U, the projective system $\mathscr{D}'(U) = ((\mathscr{D}'(V))_{V\in\mathscr{T}},(r_{WV}))$ is a presheaf on U with values in the Species of structures of locally convex topological vector spaces (TA, I, p. 42, def. 1 and p. 66, §10).