Grothendieck locally convex spaces of continuous vector valued functions
Copyright policy )Grothendieck locally convex spaces of continuous vector valued functions
Let ^{X, E) be the space of continuous functions from the completely regular Hausdorff space X into the Hausdorff locally convex space E, endowed with the compact-open topology. Our aim is to characterize the ^(X, E) spaces which have the following property: weak-star and weak sequential convergences coincide in the equicontinuous subsets of ^(X, E)'. These spaces are here called Grothendieck spaces. It is shown that in the equicontinuous subsets of E' the σ(E', E)- and β(E', ^-sequential convergences coincide, if ^(X, E) is a Grothendieck space and X contains an infinite compact subset. Conversely, if X is a G-space and E is a strict inductive limit of Frechet-Montel spaces ^(X, E) is a Grothendieck space. Therefore, it is proved that if £ is a separable Frechet space, then E is a Montel space if and only if there is an infinite compact Hausdorff X such that , E) is a Grothendieck space.
- University of Seville Spain
C(X,E) spaces, strict inductive limit of Fréchet-Montel spaces, space of continuous vector valued functions, 46E40, Spaces of vector- and operator-valued functions, Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.), Locally convex Fréchet spaces and (DF)-spaces, weak-star and weak sequential convergences, Grothendieck spaces, Vector-valued set functions, measures and integrals, compact-open topology, 46E10
C(X,E) spaces, strict inductive limit of Fréchet-Montel spaces, space of continuous vector valued functions, 46E40, Spaces of vector- and operator-valued functions, Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.), Locally convex Fréchet spaces and (DF)-spaces, weak-star and weak sequential convergences, Grothendieck spaces, Vector-valued set functions, measures and integrals, compact-open topology, 46E10
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