Corollary

Let p be a semi-norm on E. The following conditions are equivalent :

(i) p is continuous for the topology $\tau(E, F)$; (ii) every linear form f on E, such that $|f| \leq p$, comes from an element of F.

(i) $\Rightarrow$ (ii) : if p is continuous for $\tau(E, F)$, every linear form f on E such that $|f| \leq p$ is continuous for $\tau(E, F)$, hence comes from an element of F by lemma 1.

(ii) $\Rightarrow$ (i) : let $\mathscr{T}$ be the topology on E defined by the semi-norm p. If condition (ii) is satisfied, the linear forms on E which are continuous for $\mathscr{T}$ belong to F. By lemma 2 $\mathscr{T}$ is coarser than $\tau(E, F)$, hence p is continuous for $\tau(E, F)$.