Proposition 7

Let E be a semi-normed space, $E'$ its dual, B the unit ball of $E'$ and $u$ a linear mapping of $E'$ into a locally convex separated space F. If the restriction of $u$ to B endowed with the topology induced by $\sigma (E',E)$ is continuous, then $u(B)$ is a compact subset of F and $u$ is a compact linear mapping of the strong dual $E'_b$ of E into F.

Indeed the set B, endowed with the topology induced by $\sigma (E',E)$, is compact (EVT, III, p. 17, cor. 2).