The three space problem for locally bounded $F$-spaces

N. J. Kalton

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Convex-compact sets and Banach discs

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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 $σ (E^{'}, E)$ -closure of the union of countably many $σ (E^{'}, E)$ -relatively countably compacts sets, then every weakly (relatively) convex-compact set is weakly (relatively) compact.