{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/71203"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/71203","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Weak Radon-Nikdoym Sets in Dual Banach Spaces","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.","abstract_html":"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.","abstract_has_math":false,"creators":["Riddle, Lawrence Hollister"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-12-16T06:18:03Z","date_published":"2014-12-16T06:18:03Z","updated_at":"2026-07-22T22:26:04Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI8218549"],"render_values":[{"text":"(UMI)AAI8218549","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/71203","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Riddle, Lawrence Hollister"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-16T06:18:03Z","10000-01-01","1982"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/71203","(UMI)AAI8218549"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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.","Made available in DSpace on 2014-12-16T06:18:03Z (GMT). No. of bitstreams: 1 8218549.pdf: 3059165 bytes, checksum: 8a9daf91010cf356a0c6982f2c8f709f (MD5) Previous issue date: 1982","Embargo set by: Seth Robbins for item 71369 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","107 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1982."]},{"key":"dc:title","label":"Title","values":["Weak Radon-Nikdoym Sets in Dual Banach Spaces"]}]}],"canonical_facts":{"dc:creator":["Riddle, Lawrence Hollister"],"dc:date":["2014-12-16T06:18:03Z","10000-01-01","1982"],"dc:description":["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.","Made available in DSpace on 2014-12-16T06:18:03Z (GMT). No. of bitstreams: 1 8218549.pdf: 3059165 bytes, checksum: 8a9daf91010cf356a0c6982f2c8f709f (MD5) Previous issue date: 1982","Embargo set by: Seth Robbins for item 71369 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","107 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1982."],"dc:identifier":["http://hdl.handle.net/2142/71203","(UMI)AAI8218549"],"dc:subject":["Mathematics"],"dc:title":["Weak Radon-Nikdoym Sets in Dual Banach Spaces"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:26:04Z"}