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A class of topological spaces and differentiation of functions on Banach spaces. (English) Zbl 0531.46015
Let \({\mathcal A}\) and w\({\mathcal A}\) it be the classes of Asplund and weak Asplund spaces, respectively. Then, there exists a class \({\mathcal C}\) of Banach spaces such that \({\mathcal A}\subseteq {\mathcal C}\subseteq w{\mathcal A}\) and \({\mathcal C}\) is closed under the operations of
(1) taking closed subspaces,
(2) dense linear images,
(3) uncountable \(c_ 0\) sums and
(4) countable \(\ell_ 1\) sums.
Property (2) yields that WCG Banach spaces are in \({\mathcal C}\). In addition some stronger characterizations of Asplund spaces are obtained.

46B20 Geometry and structure of normed linear spaces
46B10 Duality and reflexivity in normed linear and Banach spaces
46B22 Radon-Nikodým, Kreĭn-Milman and related properties