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Title:Weak Radon-Nikdoym Sets in Dual Banach Spaces
Author(s):Riddle, Lawrence Hollister
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Abstract:The interplay between geometry, topology, measure theory and operator theory has long been evident in the study of the Radon-Nikodym 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 Radon-Nikodym 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 Radon-Nikodym property for the Pettis integral, Dunford-Pettis 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 Radon-Nikodym 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 real-valued functions formulated by Jean Bourgain in order to recognize Pettis integrable functions into dual spaces.
Issue Date:1982
Description:107 p.
Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1982.
Other Identifier(s):(UMI)AAI8218549
Date Available in IDEALS:2014-12-16
Date Deposited:1982

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