An extension of a structure theorem of Bourgain
A Banach space X is said to have the Point-of-Continuity Property (PCP) if every bounded subset of X has relatively weakly open subsets of arbitrarily small diameter. If this is true for every bounded convex subset of X then X is said to have the Convex Point-of-Continuity Property (CPCP). The main result of this paper asserts that if X fails to have CPCP, then it contains a subspace Y which also does not have CPCP, but such that Y admits a finite dimensional decomposition. Similar results for Banach spaces X having the Radon-Nikodym property or PCP, were obtained by \textit{J. Bourgain}, Stud. Math. 67, 135-148 (1980; Zbl 0434.46017) and Publ. Math. Univ. Pierre et Marie Curie, No.36 (1979).
scientific article; zbMATH DE number 4198800
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