Date of Submission

7-22-1997

Date of Award

7-22-1998

Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name

Mathematics

Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

Supervisor

Roy, Ashoke Kumar (TSMU-Kolkata; ISI)

Abstract (Summary of the Work)

In the first part of this chapter, we explain the main theme of this thesis. The second part consists of some of the notions and results used in subsequent discussions.It is a very familiar fact that a point outside a (bounded) closed convex set in a Banach space can be separated from the latter by a hyperplane. One can ask whether the separation can be effected by disjoint balls. This is a typical example of a ball separation property, study of which has become important in Banach space theory. In this thesis, we study several such properties along with some other related notions like the Asymptotic Norming Property for which, however, a ball-separation characterization is not available at present.We begin our discussion with (see the end of the section for the relevant definitions) the Asymptotic Norming Properties (ANP) of Banach spaces. The ANP was first introduced by James and Ho (JH) to show that the class of separable Banach spaces with the Radon-Nikodým Property (RNP) is larger than those isomorphic to separable duals. Three different ANP's were introduced and proved to be equivalent in separable Banach spaces. Later Ghoussoub and Mau- rey [GM] proved that in separable Banach spaces ANP's are equivalent to the RNP. Whether the two properties are equivalent in general is still an open question. Subsequently, Hu and Lin (HL1] obtained some isometric characterization of ANP's and showed that the three ANP's are equivalent in Banach spaces admit- ting a locally uniformly convex renorming, a class larger than separable Banach spaces. In dual spaces, they introduced a stronger notion called the w*-ANP, which turned out to have some nice geometric equivalents. In fact, they showed that w*-ANP-III and w*-ANP-II are respectively equivalent to Namioka-Phelps Property (referred to as the Property (**) in [NP]) and Hahn-Banach smoothness considered by Sullivan in (Su]. The latter property in turn grew out of the concept of U-subspaces intróduced by Phelps (P1). More recently, Chen and Lin (CL) have obtained some ball separation characterization of w*-ANP's which suggest similar characterization can be obtained for ANP's too. Both ANP and w*-ANP's are hereditary in nature.In the non-hereditary class of ball separation properties, i.e., properties which are not inherited by subspaces, we study nicely smooth spaces, Property (II) and the Ball Generated Property (BGP). Nicely smooth spaces were first introduced by Godefroy (G3] while the BGP by Godefroy and Kalton [GK). Property (I7), which is a natural weakening of both w*. ANP-II as well as the Mazur Intersection Property (MIP) was introduced by Chen and Lin [CL].In Chapter 2, we introduce a new ANP which lies between the strongest and the weakest ones. We denote it by ANP-II'. This new ANP has nice geometric properties and we give an example to show that it is clearly distinct from the other ANP's. We also introduce a w*-version of ANP-II'. This gives a very elegant characterization of Property (V) introduced by Sullivan in (Suj. We also study stability of this new ANP along with its w"-version.

Comments

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28843453

Control Number

ISILib-TH216

Creative Commons License

Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.

DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/2146

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