Some Modulus and Normal Structure in Banach Space

We present some sufficient conditions for which a Banach space X has normal structure in terms of the modulus of U-convexity, modulus of W*-convexity, and the coefficient R(1,X), which generalized some well-known results. Furthermore the relationship between modulus of convexity, modulus of smoothness, and Gao's constant is considered, meanwhile the exact value of Milman modulus has been obtained for some Banach space. Copyright (C) 2009 Z. Zuo and Y. Cui.