Definition 2

— Let $E$ be a locally convex space. We say that $E$ is semi-reflexive if the canonical mapping $c_E$ from $E$ into $E''$ is bijective.

This implies that $E$ is Hausdorff, and that every linear form on $E'$, which is continuous for the strong topology $\beta(E', E)$, is of the form $x' \mapsto \langle x, x' \rangle$ with $x \in E$, i.e. continuous for the weak topology $\sigma(E', E)$.