On compactness in locally convex spaces
We describe a class \({\mathcal G}\) of locally convex spaces that contains the metrizable spaces and the dual metric spaces and such that: (1) \({\mathcal G}\) is stable by the following operations: arbitrary subspaces, separated quotients, countable locally convex direct sums, countable products and completions. (2) Every space of the class \({\mathcal G}\) has metrizable precompact subsets. (3) Every space of the class \({\mathcal G}\) is weakly angelic. (4) p4 A weakly compact subset in a space of the class \({\mathcal G}\) is metrizable if, and only if, it is contained in a separable subspace. (5) A compact topological space K is such that the Banach space \(C(K)\) is weakly \(K\)-analytic if, and only if, \(K\) is homeomorphic to a weakly compact subset of a locally convex space of the class \({\mathcal G}.\) Applications to spaces of vector-valued continuous functions, to locally convex spaces with analytic duals and to the general problem of retractivity in inductive limits are also given.
spaces of vector-valued continuous functions
locally convex spaces with analytic duals
scientific article; zbMATH DE number 3975618
scientific article