Corollary 2

Let E be a semi-normed space and $\mathfrak{S}$ a set of precompact subsets of E whose union is dense in E. Let us denote by $E'_b$ the strong dual of E and by $E'_{\mathfrak{S}}$ the dual of E endowed with the $\mathfrak{S}$-topology. The space $E'_{\mathfrak{S}}$ is separated and the identical mapping of $E'_b$ into $E'_{\mathfrak{S}}$ is compact.

The space $E'_{\mathfrak{S}}$ is separated according to TG, X, p. 8, prop. 7; on the unit ball of $E'$, the $\mathfrak{S}$-topology coincides with the weak topology $\sigma (E',E)$ (EVT, III, p. 17, prop. 5), whence the corollary.