On weakly locally nearly uniformly convex Banach spaces
by
Sergio Falcon
Department of Mathematics. University of Las Palmas de G.C. Spain
Coauthors: Ignacio Cabrera (University of Las Palmas de G.C.), Kishin Sadarangani (University of Las Palmas de G.C.)

In recent years there have appeared some papers containing generalizations of the concept of convexity with help of the measures of noncompactness. The weak version of this convexity uses the De Blasi measure of weak noncompactness. In this paper we show that every Banach space with a weakly locally nearly uniformly convex norm has an equivalent locally nearly uniformly convex norm. This result is the noncompact translation of a recent result proved by Moltó, Orihuela, Troyanski, and Valdivia in the classic case.
We prove two theorems:
Theorem 1: A Banach space E is weakly locally nearly uniformly convex if and only if it is reflexive.
Theorem 2: If a Banach space E is weakly locally nearly uniformly convex then E has a locally nearly uniformly convex equivalent norm.

(T)

Date received: March 24, 2000