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Weak holomorphy and other weak properties. (English) Zbl 1077.46030
In the present short note, the authors show with a very simple proof that, for a vector function with the property that the range of each compact set is contained in an absolutely convex weakly compact set, it suffices to check holomorphy on the elements of a separating subset of the dual of the (locally complete locally convex) range space. That is, the authors have a somewhat stronger hypothesis on the mapping to start with and do not recover Grosse-Erdmann’s result unless the range space is semireflexive. Moreover, contrary to Grosse-Erdmann, the authors also make use of the Dunford-Grothendieck theorem on the equivalence of holomorphy and weak holomorphy.

46E40Spaces of vector- and operator-valued functions
46A20Duality theory of topological linear spaces
46A32Spaces of linear operators; topological tensor products; approximation properties
46E10Topological linear spaces of continuous, differentiable or analytic functions