Corollary 2

— Every equicontinuous subset of the dual $E'$ of E is relatively compact for the weak topology $\sigma(E', E)$ on $E'$ (III, p. 14, Example 4).

For, if H is an equicontinuous subset of $E'$, $\sup_{u \in H} |u|$ is a continuous semi-norm on E; in particular, for every $x \in E$, the set $H(x)$ is bounded, hence relatively compact in the field of scalars.