Local Embeddings Of Banach Spaces

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.