## Doctoral Theses

### Intersection Properties of Balls in Banach Spaces and Related Topics.

2-22-2003

2-22-2004

#### Institute Name (Publisher)

Indian Statistical Institute

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

Mathematics

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

#### Abstract (Summary of the Work)

In the first part of this chapter, we explain in general terms the background and the main theme of this thesis and provide a chapter-wise summary of its principal results. In the second part, we introduce some notations and preliminaries that will be used in the subsequent chapters.As a prototype of the properties we will study in this thesis, let us call a closed linear subspace Y of a Banach space X a (P)-subspace of X if Y has a certain property P as a subspace of X. If a Banach space X, in its canonical embedding, is a (P)-subspace of its bidual X**, that often endows X with a rich geometric structure. M-embedded spaces-spaces that are M-ideals in their biduals-is a case in point. See [37, Chapter 3] for properties of such spaces. Other examples of such properties include Hahn-Banach smooth spaces, 1-complemented subspaces of the bidual etc.On the other hand. many geometric properties of a Banach space X, in some equivalent formulation, identifies X as a (P)-subspace of X**. It is often a more interesting exercise to study the general property of being a (P)-subspace of X itself. This very often yields a better understanding of the original geometric property to. We take this approach in studying some geometric properties considered in the literature. As one would expect, in doing so, we often need an algebraic, in contrast to the topological approach of the original treatments and have to develop proper tools for this analysis.Another common feature of the properties we propose to study is that the property of being a (P)-subspace is formulated in terms of closed balls in X with centres in the subspace Y, often as an intersection property of families of such balls. In the X in X**set-up, such properties have been studied quite extensively. For example, we may mention the work of Linden- strauss on Pi-spaces and L1-predual spaces in [48], the works of Godefroy [32], Godefroy and Kalton [33], Godefroy and Saphar [34 and many other authors in the study of norm 1 complementability of X in X**, existence of unique isometric predual of a dual Banach space, the ball topology and its applications, etc.With this approach, we start with the study of the finite infinite inter- section property (IPf,00) studied by Godefroy in (32] and by Godefroy and Kalton in [33]. From an equivalent formulation of the IPf0, we isolate the corresponding subspace property as followsDefinition 1.1.1. A closed linear subspace Y of a Banach space X is said to be an almost constrained (AC) subspace of X if any family of closed balls centred at points of Y that intersects in X also intersects in Y.Clearly, a constrained (that is, 1-complemented) subspace is almost con- strained. It is well known that all dual spaces and their constrained sub- spaces have IPf.00. One of the main results in [32] is that if a Banach space X does not contain an isomorphic copy of l1 and has the IPt00, then X is a dual space and it is complemented in X** by a unique norm 1 pro- jection, which in turn ensures that X has a unique isometric predual.

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:28842919

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