Remark 2

Let $\mathscr{T}_1$ and $\mathscr{T}_2$ be two locally convex topologies on E, compatible with the duality between E and $E'$, $\mathscr{T}_1$ being finer than $\mathscr{T}_2$. Then every neighbourhood of 0 for $\mathscr{T}_1$, which is convex and closed for $\mathscr{T}_1$ is closed for $\mathscr{T}_2$ by prop. 1 of IV, p. 1. Consequently (GT, II, § 3, No. 3, corollary) every subset of E which is complete for $\mathscr{T}_2$ is so for $\mathscr{T}_1$ also.

In particular, every subset of E which is complete for the weakened topology is complete for the initial topology, and every subset of E complete for the initial topology is so for the Mackey topology. If E is quasi-complete for the weakened topology, it is so for every topology compatible with the duality between E and $E'$. If it is quasi-complete for the initial topology, it is so for the Mackey topology.