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Countably determined locally convex spaces. (English) Zbl 0765.46004

The paper is an application of some results on countably determined and \(K\)-analytic topological spaces to locally convex spaces (l.c.s.) with the properties of quasi-completeness, semireflexivity, etc. Thus the technique appeals to considerations from general topology. Also, the natural case of a functional (l.c.s.) space of vector-valued continuous functions with compact-open topology is considered with the goal to study the question of existence of metrizable compact subsets in the functional space; the authors formulate a corresponding result on metrizability for uniform spaces. All the l.c.s. considered in the paper belong to class \(G\), defined by the authors as follows: a l.c.s. \(E\) belongs to class \(G\), if there exists a naturally by \(N\) ordered set in the dual space \(E^*\), covering \(E^*\) and such that every countable subset of the set is equicontinuous. This class, in particular, includes barrelled l.c.s. with \(K\)-analytic weak-\(^*\) dual spaces (the definition is a natural generalization of the corresponding definition of quasi-\(LB\)-spaces of M. Valdivia).
The definition enables the authors to apply a topological technique of countably based filters. The authors characterize l.c.s. which are quasi- complete in the Mackey topology and countably-determined in terms of covering by compact sets generated by an appropriate separable metric space; this is a consequence of a variant of characterization of countably determined regular topological spaces (see a series of works of M. Talagrand on \(K\)-analytic spaces related to the subject). The authors prove also that if a complete quasi-\(LB\) space [see M. Valdivia, J. Lond. Math. Soc., II. Ser. 35, 149-168 (1987; Zbl 0625.46006)] contains a subset \(S\) which is countably determined in a weak topology, and such that the linear hull of \(S\) is fast-dense in \(E\) (= modulo sequences that are Mackey-convergent in Banach disc), then \(E\) is weakly countably determined.
Reviewer: S.Berger (Emek)

MSC:

46A50 Compactness in topological linear spaces; angelic spaces, etc.
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
46A17 Bornologies and related structures; Mackey convergence, etc.
46A03 General theory of locally convex spaces
46A13 Spaces defined by inductive or projective limits (LB, LF, etc.)

Citations:

Zbl 0625.46006
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