Abstract
LetE andF be locally convex topological vector spaces. A holomorphic mapf: E→F is defined to be an Asplund map if it takes the separable subsets of a neighbourhood of eacha∈E into absolutely convex weakly metrisable subsets ofF; a Banach space is an Asplund space if and only if its identity map has this property. We show that a continuous linear map from a quasinormable locally convex spaceE into a Banach spaceF is an Asplund map if and only if it factors through an Asplund space. IfE andF are both Banach spaces, then a holomorphic mapf: E→F is an Asplund map if and only if its derivative maps\(\hat d^k f(a)\) factor through Asplund spaces for eacha∈E. This is true if and only if such a factorisation holds ata=0.
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Part of this research was done during a visit to the University of Namibia, whose financial support is gratefully acknowledged
This article was processed by the author using the Springer-Verlag TEX mamath macro package 1990
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Robertson, N. Asplund operators and holomorphic maps. Manuscripta Math 75, 25–34 (1992). https://doi.org/10.1007/BF02567068
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DOI: https://doi.org/10.1007/BF02567068