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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Geometrical implications of certain infinite-dimensional decompositions

Authors: N. Ghoussoub, B. Maurey and W. Schachermayer
Journal: Trans. Amer. Math. Soc. 317 (1990), 541-584
MSC: Primary 46B20; Secondary 46B15, 46B22
MathSciNet review: 962281
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Abstract: We investigate the connections between the "global" structure of a Banach space (i.e. the existence of certain finite and infinite dimensional decompositions) and the geometrical properties of the closed convex bounded subsets of such a space (i.e. the existence of extremal and other topologically distinguished points). The global structures of various--supposedly pathological-- examples of Banach spaces constructed by R. C. James turn out to be more "universal" than expected. For instance James-tree-type (resp. James-matrix-type) decompositions characterize Banach spaces with the Point of Continuity Property (resp. the Radon-Nikodým Property). Moreover, the Convex Point of Continuity Property is stable under the formation of James-infinitely branching tree-type "sums" of infinite dimensional factors. We also give several counterexamples to various questions relating some topological and geometrical concepts in Banach spaces.

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Keywords: James-type decompositions, boundedly complete and $ {l^1}$-skipped blocking decompositions, Radon-Nikodým and Point of Continuity Properties, strongly regular spaces
Article copyright: © Copyright 1990 American Mathematical Society