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Title:  Weak RadonNikdoym Sets in Dual Banach Spaces 
Author(s):  Riddle, Lawrence Hollister 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Mathematics 
Abstract:  The interplay between geometry, topology, measure theory and operator theory has long been evident in the study of the RadonNikodym property. Recently results of substantial interest in the structure of Banach spaces have been obtained by localizing these ideas to individual subsets. The study of the RadonNikodym property for subsets of Banach spaces can be thought of as the study of subsets of Banach spaces whose structural properties mimic those of the unit ball of a separable dual space. In this thesis I initiate the study of geometric, topological, measure theoretic and operator theoretic characterizations of convex weak*compact subsets of dual Banach spaces whose structural properties mimic those of the unit ball of the dual of a space that contains no copy of the sequence space l(,1). These sets are described in terms of the RadonNikodym property for the Pettis integral, DunfordPettis operators, points of weak*continuity and universal weak*measurability of linear functionals in the second dual, extreme points, Rademacher trees, dentability and convergent martingales. By and large the work is based on a factorization theorem that says that an operator T : X (>) Y factors through a Banach space containing no copy of l(,1) if and only if the adjoint operator T* maps the unit ball of Y* into a set with the RadonNikodym property for the Pettis integral. Also included in the thesis are several sufficient conditions for Pettis integrability. Using a deep theorem of Bourgain, Fremlin and Talagrand, I show that every bounded universally scalarly measurable function from a compact Hausdorff space into the dual of a separable space is universally Pettis integrable. In addition, I use a property of families of realvalued functions formulated by Jean Bourgain in order to recognize Pettis integrable functions into dual spaces. 
Issue Date:  1982 
Type:  Text 
Description:  107 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 1982. 
URI:  http://hdl.handle.net/2142/71203 
Other Identifier(s):  (UMI)AAI8218549 
Date Available in IDEALS:  20141216 
Date Deposited:  1982 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois