Normal Structure and Some Inequalities of Geometric Parameters in Banach Space X and X ∗

Abstract

Let $X$ and $X^{\ast }$ be a Banach space and its dual, respectively. In this paper, we study the relations between modulus of $W_{1X^{\ast }}\left(\epsilon \right)$ and modulus ${\zeta }_{X^{\ast }}\left(\epsilon \right)$ in $X^{\ast }$ and normal structure in $X$ , respectively. Among other results, we proved either $W_{1X^{\ast }}\left(\epsilon \right)<\epsilon /2$ , for any $0\le \epsilon \le 2$ , or ${\zeta }_{X^{\ast }}\left(\epsilon \right)<1+\epsilon$ , for any $0\le \epsilon \le 1$ , implies both $X$ and its dual $X^{\ast }$ have uniform normal structure.