The Lifting Problem Of Some Geometric Properties Of Banach Spaces In The Sequential Space

In the recent years, the studies on the geometric theory of Banach space have achieved rapid developments. Up to now, the explorations on convexity, smoothness, differentiability, roughness, drop-property, and convergence of general Banach spaces 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 {E_i}. Therefore, in this paper, I mainly explore theconvexity, smoothness, and convergence of the Banach sequential spaces l~p(E_i) and ces_p(E_i), and obtain some good results.This paper consists of five parts.Chapter One The basic definitions and necessary lemmas posed in this paper.Chapter Two The lifting problem about property (C-K) in l~p(E_i) and ces_p(E_i). In this chapter, the convex property (C-K), which is weaker than LkR, is lifted to the sequential spaces l~p(E_i) and ces_p(E_i). In addition, one necessary and sufficient condition for the sequential spaces ces_p(E) to be CLwR spaces (resp wCLwR spaces) is obtained.Chapter Three In this chapter, a new geometric property, a smooth property, that is strictly stronger than the smooth property (S-K), is introduced.So it is the generalization of property (S-K). Moreover, some properties on (Sk-K) are obtained, and lifted to the sequential space l"(E,). Besides, thelifting problem about some smooth properties such as LkS, wLkS, CLï¿¡S, and wCLkSia /'(ï¿¡;) are discussed, and they are lifted to l"(E() as well.Chapter Four In this chapter, the chapter of the smooth point of cesp(E) is discussed, and one necessary and sufficient condition for x=(xR)ecesp(E) (xk*Q and H = l) to be a smooth point of cesp(E) is obtained.Chapter Five The lifting problem about the convergent property (u)in cesp(EH). In this chapter, property (u) is lifted to the sequential space cesp(En).