Results Regarding Precompactness

Proposition 1: Let $(E, F)$ be a dual pair and let $A \subseteq E$. If $A$ is $\sigma(E, F)$-weakly bounded then $A$ is weakly precompact.

By "weakly precompact" we mean that $A$ is precompact when $E$ is equipped with the weak topology on $E$ determined by $F$.

Proposition 2: Let $E$ be a Hausdorff locally convex topological vector space and let $A \subseteq E$. Then: (1) If $A$ is precompact then $\mathrm{conv}(A)$ is precompact. (2) If $A$ is precompact then $\mathrm{abs \: conv}(A)$ is precompact. (3) If $A$ is precompact then $\overline{\mathrm{abs \: conv}(A)}$ is precompact.