weakly compact sets; convex-compact sets; Banach discs
Every relatively convex-compact convex subset of a locally convex space is contained in a Banach disc. Moreover, an upper bound for the class of sets which are contained in a Banach disc is presented. If the topological dual $E'$ of a locally convex space $E$ is the $\sigma (E',E)$-closure of the union of countably many $\sigma (E',E)$-relatively countably compacts sets, then every weakly (relatively) convex-compact set is weakly (relatively) compact.
 Köthe, G.: Topological Vector Spaces I
. Springer-Verlag (1969). MR 0248498
 Pták, V.: A combinatorial lemma on the existence of convex means and its applications to weak compactness
. Proc. Symp. Pure Math. VII (Convexity 1963) 437-450. MR 0161128