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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 <IMG WIDTH="21" HEIGHT="23" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="${l^1}$">-skipped blocking decompositions, Radon-Nikod&#253;m and Point of Continuity Properties, strongly regular spaces
Article copyright: © Copyright 1990 American Mathematical Society