【作者】 阿拉腾苏布德；

【导师】 苏雅拉图；

【作者基本信息】 内蒙古师范大学 ， 基础数学， 2005， 硕士

【摘要】 近年来,Banach空间几何理论研究得到迅速的发展,到目前为止,一般Banach空间的凸性、光滑性、可微性、粗性、滴性、收敛性的研究已比较完善,但是由一列Banach空间{E_i}所产生的Banach序列空间l~p(E_i)和ces_p(E_i)的凸性、光滑性、可微性、粗性、滴性、收敛性的研究还不是很完善。本文研究了Banach序列空间l~p(E_i)和ces_p(E_i)的某些凸性、光滑性和收敛性,得到了较好的结果,全文共分为五章。 第一章:预备知识 第二章:本章中讨论了弱于LkR(局部完全k-凸)的凸性质(C-K)在两类序列空间l~p(E_i)和ces_p(E)中的提升问题,并把(C-K)性质提升到了序列空间l~p(E_i)和ces_p(E)。此外,给出了ces_p(E) (1<p<∞)为紧局部完全w凸(弱紧局部完全w凸)的判据。 第三章:本章中引入了一种新的几何性质(Sk-K),它是严格强于光滑性质(S-K)的一种光滑性质,并且是(S-K)性质的推广。本章中笔者得到了(Sk-K)的某些性质,并把(Sk-K)提升到了序列空间l~p(E_i)。此外,讨论了某些光滑性质LkS(wLkS、CLkS、wCLkS)在l~p(E_i)中的提升问题,并把它们提升到了序列空间l~p(E_i)。 第四章:本章中讨论了ces_p(E)的光滑点的特征,得到了x=(x_k)∈ces_p(E)(x_k≠0且‖x‖=1)成为ces_p(E)的光滑点的充分必要条件。 第五章:本章中讨论了收敛性质(u)在ces_p(E_n)中的提升问题,并把(u)性质提升到了序列空间ces_p(E_n)更多还原

【Abstract】 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, CL￡S, and wCLkSia /’(￡;) 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).更多还原