Proposition 5

— Let E, F be two topological vector spaces over a non-discrete valued division ring K; suppose that the topology of E (resp. F) is defined by a set of semi-norms $\Gamma$ (resp. $\Gamma'$). Let H be a set of linear mappings of E in F. The following conditions are equivalent : a) H is equicontinuous.

b) For every semi-norm $q \in \Gamma'$, there exists a finite family $(p_i)_{1 \leq i \leq n}$ of semi-norms belonging to $\Gamma$ and a number $a > 0$ such that, for all $x \in E$ and all $u \in H$,

c) For every semi-norm $q \in \Gamma'$, the mapping $\sup_{u \in H} (q \circ u)$ is a continuous semi-norm on $E$.