Journal of Convex Analysis 12 (2005), No. 1, 159--172
Copyright Heldermann Verlag 2005
Strong Martingale Type and Uniform Smoothness
Jörg Wenzel
Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa
wenzel@minet.uni-jena.de
We introduce stronger versions of the usual notions of martingale type p ≤ 2 and cotype q ≥ 2 of a Banach space X and show that these concepts are equivalent to uniform p-smoothness and q-convexity, respectively. All these are metric concepts, so they depend on the particular norm in X.
These concepts allow us to get some more insight into the fine line between X being isomorphic to a uniformly p-smooth space or being uniformly p-smooth itself.
Instead of looking at Banach spaces, we consider linear operators between Banach spaces right away. The situation of a Banach space X can be rediscovered from this by considering the identity map of X.
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