with $f,\phi\in C^\infty(P)$.

We take the space $C^\infty(P)$ (sometimes denoted $\mathscr E$) of smooth functions on $P$ to be endowed with the topology of convergence on compact sets, which is induced by the family of seminorms $|\cdot|_{K,k}$ where $K\subset P$ is a compact set, $k{\geqslant}0$ is an integer, and $$\label{eq:seminorms}
  |f|_{K,k}=\sum_{|I|{\leqslant}k}\sup_{p\in K}|\partial^If(p)|,\quad f\in C^\infty(P).$$ The subspace $C^\infty_c(P)\subseteq C^\infty(P)$ (sometimes denoted $\mathscr D$) of compactly supported smooth functions inherits the topology from $C^\infty(P)$.

Denote by $\mathscr D'$ the space of distributions on $P$, and by $\mathscr E'$ the space of compactly-supported distributions on $P$. These are endowed with the weak\* topology with respect to $C^\infty_c(P)$ and $C^\infty(P)$, respectively. We further write $\mathscr E'=\cup_n\mathscr E'^n$, where $\mathscr E'^n$ is the set of compactly-supported distributions of order ${\leqslant}n$ for $n\in{\mathbb{N}}\cup\{0\}$.

For a positive integer $m$ and a compact set $K\subset P$, let $|\cdot|_{K,m}$ be the norm given in , and let $|\cdot|'_{K,m}$ be the dual seminorm given by $$\label{eq:dualseminorms}
  |\xi|'_{K,m}=\sup_{|g|_{K,m}{\leqslant}1} |\langle g,\xi\rangle|,\qquad\xi\in\mathscr E',$$ where the supremum is taken over all functions $g\in C^\infty_c(P)$ such that $|g|_{K,m}{\leqslant}1$.

For the proof of Theorem \[thm:families\] we will need this lemma, which is proved below.

\[lem:boundednormability\] Let $n$ be a positive integer, $\gamma>0$, and $U$ a bounded 