The Study On The Some Geometric Properties Of Banach Space And The Lifting Problems

In the recent years, the studies on the geometric theory of Banach space have achieved rapid development. Up to now, the exploration on convexity, smoothness, differentiability, roughness, drop-property, and convergence of general Banach space have been relatively perfect while they haven't been very perfect in Banach sequential spaces, which is generated by a series of the Banach sequential spaces { }X i, and some new k-smoothness and k-convexity has not introduced in Banach spaces yet. Therefore, in his paper, I mainly explore the two kinds of k-smoothness in Banach spaces and some lifting problems in sequential spaces ( )ces p X i. In the process of my study, I have obtained some productive results.This paper consists of two parts. Chaper One Two kinds of k-smoothness in Banach spacesIn this chaper , we introduce the daul notions of fully k-convex spaces and k-nearly uniformly convex spaces.i.e., fully k-smooth spaces and k-nearly uniformly smooth spaces. We prove that both fully k-smooth spaces and k-nearly uniformly smooth spaces are reflexive, and obtained the some characters and properties of them. Also, we prove that fully k-smooth (resp.k-nearly uniformly smooth) spaces implies fully (k+1)- smooth spaces(resp. (k+1)-nearly uniformly smooth). However, the converse inclusion is not necessarily true.Chaper Two The lifting of some geometric properties on sequence space ces p ( X k)In this chaper, we mainly study the problems about lifting of some geometric property from Banach space X k to sequence spaces ces p ( X k).