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On compactness in locally convex spaces. (English) Zbl 0604.46011

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.

MSC:

46A50 Compactness in topological linear spaces; angelic spaces, etc.
46A04 Locally convex Fréchet spaces and (DF)-spaces
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
54E35 Metric spaces, metrizability
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References:

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