【作者】 程庆进；

【导师】 程立新；

【作者基本信息】 厦门大学 ， 基础数学， 2007， 博士

【摘要】 Banach空间的局部嵌入问题与研究空间结构、粗嵌入及空间上算子的因子分解有密切关系.本文目的是引入超弱紧集的概念并研究下面这几类局部集合的嵌入问题及其应用:R.N.P集、controllable R.N.P集[1]、controllableP.C.P集[1]、弱紧集及超弱紧集.全文围绕着“超弱紧集的嵌入”这一中心问题展开研究,共分密切相关的七章:第一章简要回顾了空间嵌入研究的发展概况并给出了本文研究的目的和意义.第二章通过改进著名的Davis-Figiel-Johnson-Pelzyn′ski引理[2]得到Banach空间的每个弱紧集均可一致嵌入进某个自反Banach空间,这从本质上改进了文献[2]的结果.作为它的应用我们建立了弱紧集版的Odell-Schlumprech定理[3]和Ha′jek-Johanis定理[4].另外还得到每个controllableP.C.P集以及每个controllable R.N.P集均可分别一致嵌入进某个P.C.P空间和R.N.P空间.第三章利用推广的有限表示的概念,引入了一个可视作超自反Banach空间概念的推广和局部化的概念―――超弱紧集,主要证明了一个Banach空间是超自反的等价于其闭单位球是超弱紧的.第四章本章目的是去得到超弱紧集的下面一个特征:一个有界闭凸集是超弱紧的当且仅当其不具有有限树性质.第五章在超弱紧集有限树特征的基础上,通过推广和开发En?o再赋范定理[5]证明的一系列方法和技巧,最后我们得到了超弱紧集的两个凸函数特征:ε-一致凸函数特征和一致凸函数特征.第六章首先给出了超弱紧集版的Grothendieck引理,然后通过第二章改进的Davis-Figiel-Johnson-Pelzyn′ski引理和上章建立的超弱紧集的凸函数特征,最后证明了Banach空间中的每个超弱紧集均可一致嵌入进某个自反且相对一致凸空间.作为它的应用给出了超弱紧集与超B.S.P的关系.第七章研究了超弱紧集在再赋范成为一致凸集的几何性质.并针对历史上出现的几类一致凸集给出了几个注记.更多还原

【Abstract】 Local embedding problems of Banach spaces are closely linked to the study of thestructure of spaces, coarse embeddings and operator factorization. The purpose of thisthesis is to introduce a notion of super-weakly compact sets and focus on investigatingtheir embedding problems and applications of the following several classes of sets:R.N.P set, controllable P.C.P set [1], controllable R.N.P set [1], weakly compact setand super-weakly compact set. The thesis centers around a central problem”super-weakly compact sets, embeddings”to begin with, and it consists of closely relatedseven chapters.Chapter 1 presents a survey of the study of embedding problems of Banach spacesand gives the aim and meaning of the study of the thesis.Chapter 2, by improving famous Davis-Figiel-Johnson-Pelzyn′ski Lemma, obtainsthat every weakly compact subsets of Banach spaces can be uniformly embeded intosome re?exive Banach space, which is an essential improvement to the result of [2].As its application, builds weakly compact set versions of Odell-Schlumprech theoremand Ha′jek-Johanis theorem. Otherwise, obtains every controllable P.C.P set and everycontrollable R.N.P set can be uniformly embeded into some P.C.P space and R.N.Pspace, respectively.Chapter 3 introduces a notion of super-weakly compact set of Banach spaces interms of a generalized notion of finite representability, which is a generalized andlocalized setting of super-re?exive Banach spaces, and mainly shows that a Banachspace is super-re?exive if and only its closed unit ball is super-weakly compact.Chapter 4 intends to build the following characterization of super-weakly compactsets: a bounded and closed convex set is super-weakly compact if and only it does nothave finite tree property.Chapter 5, on the basic of finite tree characterization of super-weakly compactsets, by extending and developing a series of methods and techniques in the proof ofEn?o,s renorming theorem, finally establishes characterizations of two convex func-tions of super-weakly compact sets: characterization ofε-uniformly convex functionand characterization of uniformly convex function. Chapter 6 first verifies that Grothendieck,s Lemma for weakly compact setsis again valid for super-weakly compact sets, then by the improved Davis-Figiel-Johnson-Pelzyn′ski Lemma in Chapter 2 and characterizations of convex functions ofsuper-weakly compact sets obtained in above chapter, finally gives that every super-weakly compact set can be uniformly embeded into some re?exive and relatively uni-formly convex space. As its application, establishes a relation between super-weaklycompact sets and super-Banach-sakes property.Chapter 7 investigates some geometric properties of super-weakly compact sets un-der renorming uniformly convex norm, and gives some remarks about several classesof uniformly convex sets which have appeared in the literature.更多还原