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arXiv · 1910.11962

Nondentable Sets in Banach Spaces

S. J. Dilworth·Chris Gartland·Denka Kutzarova·N. Lovasoa Randrianarivony
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Abstract

In his study of the Radon Nikod\'ym property of Banach spaces, Bourgain showed (among other things) that in any closed, bounded, convex set $A$ that is nondentable, one can find a separated, weakly closed bush. In this note, we prove a generalization of Bourgain's result: in any bounded, nondentable set $A$ (not necessarily closed or convex) one can find a separated, weakly closed approximate bush. Similarly, we obtain as corollaries the existence of $A$-valued quasimartingales with sharply divergent behavior.

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S. J. Dilworth, Chris Gartland, Denka Kutzarova, N. Lovasoa Randrianarivony. 2019-10-25. Nondentable Sets in Banach Spaces. https://arxiv.org/abs/1910.11962

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