Some Geometric Properties Of Banach Space

Smoothess as the dual conception of convexity was introduced, at the same time, the promotion of smoothess also obtained a rapid development. The research of the geometric constant to depict the geometric property of Banach space has a very important value, the specific value rang of geometric constants for Banach space determines the existence of some geometrical properties, so it is a hot topic to calculate the value of constant and modulus for domestic and foreign scholars in concrete Banach space.The some equivalence conditions of kUS Banach space is studied, mean- -while some basic properties and the conception of L ? ksspace are provided. At last, with the help of absolute normalized norms, the precise values of constant for some classical Banach space are given, some sufficient conditions which imply normal structure are obtained. The contents of the paper are organized as follow:In the first part, the conception of local fully k smooth space is introcuced, and some basic properties of local fully k smooth space and two new features of LUS space are given. Namely, the relationship between LUS and L ? kS and LkUS are depicted. Fianlly, the some equivalence conditions of kUS Ban- ach space are given.In the second part, with the help of absolute normalized norms, the precise value of constantC_Z(X) and T ( X ) in some concrete Banach spaces are given. Finally, some sufficient conditions are given by the constant T ( X ), which imply Banach space has normal structure.