Remark 2

Let K be a convex subset of F which is compact for the weak topology $\sigma(F, E)$ and $\mu$ a positive measure on K. Put

for all $x \in E$. It is immediate that p is a semi-norm. Moreover, for every $x \in E$, the relation « $|B(x, y)| \leq 1$ for all $y \in K$ » implies that $p(x) \leq \mu(K)$. This proves that the semi-norm p on E is continuous for the Mackey topology $\tau(E, F)$. \*