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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.
[1] Day, M. M.: Normed Linear Spaces. Spriger-Verlag (1973). MR 0344849 | Zbl 0268.46013
[2] Floret, K.: Weakly compact sets. Lecture Notes in Math., Springer-Verlag 801 (1980). MR 0576235 | Zbl 0437.46006
[3] Grothendieck, A.: Critères de compacité dans les espaces fonctionnels généraux. Amer. J. Math. 74 (1952), 168-186. DOI 10.2307/2372076 | MR 0047313 | Zbl 0046.11702
[4] Köthe, G.: Topological Vector Spaces I. Springer-Verlag (1969). MR 0248498
[5] 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
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