Corollary

— *Let E be a locally convex semi-reflexive space. Every closed vector subspace M of E is semi-reflexive; moreover, the strong topology on $E'/M^\circ$ (considered as the dual of M) is the quotient of the strong topology on E'.

Let B be a bounded subset of M. Since B is bounded in E, and the weakened topology $\sigma(M, M')$ on M is induced by $\sigma(E, E')$ (IV, p. 10, prop. 11), the closure of B in M endowed with $\sigma(M, M')$ is compact. Hence, by th. 1, M is semi-reflexive. The last assertion of the corollary follows from prop. 10 of IV, p. 9, applied to the set $\mathscr{S}$ of all closed, convex and bounded subsets of E.