Abstract
In his study of the Radon-Nikodým 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.
Original language | English (US) |
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Pages (from-to) | 31-40 |
Number of pages | 10 |
Journal | Journal of Convex Analysis |
Volume | 28 |
Issue number | 1 |
State | Published - 2021 |
Keywords
- Convex sets
- Dentable sets in normed spaces
- Extreme points
- Martingale convergence
- Radon-Nikodým property
ASJC Scopus subject areas
- Analysis
- General Mathematics