On the modulus of U-convexity

Abstract

We prove that the moduli of U-convexity, introduced by Gao (1995), of the ultrapower X˜ of a Banach space X and of X itself coincide whenever X is super-reflexive. As a consequence, some known results have been proved and improved. More precisely, we prove that uX(1)>0 implies that both X and the dual space X∗ of X have uniform normal structure and hence the “worth” property in Corollary 7 of Mazcuñán-Navarro (2003) can be discarded.