Files in this item

FilesDescriptionFormat

application/pdf

8218549.pdf (3MB)
(no description provided)PDF

Description

 Title: Weak Radon-Nikdoym Sets in Dual Banach Spaces Author(s): Riddle, Lawrence Hollister Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign 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 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 Type: Text Description: 107 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1982. URI: http://hdl.handle.net/2142/71203 Other Identifier(s): (UMI)AAI8218549 Date Available in IDEALS: 2014-12-16 Date Deposited: 1982
﻿