Corollary 1

Let E be a semi-normed space of countable type, let $E'$ be its dual and let B be the unit ball of $E'$. Let $u$ be a linear mapping of $E'$ into a locally convex separated space F. Suppose that for every sequence $(x_n)_{n\in\mathbf{N}}$ of elements of B which converges for $\sigma (E',E)$ to 0, the sequence $(u(x_n))_{n\in\mathbf{N}}$ converges to 0 in F. 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 topology induced on B by $\sigma (E',E)$ is metrisable (EVT, III, p. 19, cor. 2). The assumption of the corollary therefore implies that the restriction of $u$ to B is a continuous mapping of B into F, when B is endowed with the topology $\sigma (E',E)$, and the corollary follows from the preceding proposition.