Remark

— Suppose in addition, that there exists a countable total set in $E$; then every equicontinuous subset of $E'$ is metrizable for the topology $\sigma(E', E)$ (III, p. 18, prop. 6); therefore to verify that a linear form $u$ on $E'$ is weakly continuous, it is enough to verify that for every equicontinuous sequence $(x'_n)$ in $E'$ which converges to 0 for $\sigma(E', E)$, we have $\lim_{n \to \infty} u(x'_n) = 0$.