{"id":{"repo_id":"cape-town","oai_identifier":"oai:open.uct.ac.za:11427/21966"},"canonical_url":"https://search.dev.ndltd.org/etd/cape-town/oai:open.uct.ac.za:11427/21966","repository":{"repo_id":"cape-town","name":"University of Cape Town","base_url":"https://open.uct.ac.za/oai/request"},"display":{"title":"Separability and metrisability in locally convex spaces","abstract":"This thesis is devoted to a study of the relationship between separability and metrisability in the context of locally convex spaces. The duality between sep- arability and weak*-metrisability does not carry over to non-metrisable locally convex spaces; the best that can be said in this case is that the equicontinuous subsets in the dual of a separable locally convex space are weak*-metrisable. To get around this difficulty, we often prefer to use the idea of separability by seminorm: a locally convex space E is separable by seminorm if and only if the equicontinuous subsets of its dual are weak*-metrisable. On any locally convex space E there is a finest topology Tχ which is coarser than the given topology and which makes E separable by seminorm. A question that arises is under what conditions a space E is Tχ-complete. In trying to answer this question, we are led to an intriguing binary relation which G.A. Edgar originally defined on the class of Banach spaces. In the first two Chapters of this thesis, we show that many of the results in Edgar's paper can be expressed in terms of the completeness of a space with respect to various topologies.","abstract_html":"This thesis is devoted to a study of the relationship between separability and metrisability in the context of locally convex spaces. The duality between sep- arability and weak*-metrisability does not carry over to non-metrisable locally convex spaces; the best that can be said in this case is that the equicontinuous subsets in the dual of a separable locally convex space are weak*-metrisable. To get around this difficulty, we often prefer to use the idea of separability by seminorm: a locally convex space E is separable by seminorm if and only if the equicontinuous subsets of its dual are weak*-metrisable. On any locally convex space E there is a finest topology Tχ which is coarser than the given topology and which makes E separable by seminorm. A question that arises is under what conditions a space E is Tχ-complete. In trying to answer this question, we are led to an intriguing binary relation which G.A. Edgar originally defined on the class of Banach spaces. In the first two Chapters of this thesis, we show that many of the results in Edgar's paper can be expressed in terms of the completeness of a space with respect to various topologies.","abstract_has_math":false,"creators":["Robertson, Neill Raymond Charles"],"institution":"Department of Mathematics and Applied Mathematics","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Webb, John H"],"committee_chairs":[],"committee_members":[],"year":1991,"date_issued":"1991","date_published":"1991","updated_at":"2026-07-22T22:23:09Z","subjects":[],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/11427/21966","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Webb, John H"]},{"key":"dc:creator","label":"Author","values":["Robertson, Neill Raymond Charles"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2016-09-28T18:59:39Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2016-09-28T18:59:39Z"]},{"key":"dc:date.issued","label":"Date","values":["1991"]},{"key":"dc:publisher.department","label":"Dc Publisher Department","values":["Department of Mathematics and Applied Mathematics"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Cape Town"]},{"key":"dc:type","label":"Dc Type","values":["Doctoral Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["Doctoral"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["PhD"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["eng"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/11427/21966"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Bibliography: pages 58-61."]},{"key":"dc:description.abstract","label":"Abstract","values":["This thesis is devoted to a study of the relationship between separability and metrisability in the context of locally convex spaces. The duality between sep- arability and weak*-metrisability does not carry over to non-metrisable locally convex spaces; the best that can be said in this case is that the equicontinuous subsets in the dual of a separable locally convex space are weak*-metrisable. To get around this difficulty, we often prefer to use the idea of separability by seminorm: a locally convex space E is separable by seminorm if and only if the equicontinuous subsets of its dual are weak*-metrisable. On any locally convex space E there is a finest topology Tχ which is coarser than the given topology and which makes E separable by seminorm. A question that arises is under what conditions a space E is Tχ-complete. In trying to answer this question, we are led to an intriguing binary relation which G.A. Edgar originally defined on the class of Banach spaces. In the first two Chapters of this thesis, we show that many of the results in Edgar's paper can be expressed in terms of the completeness of a space with respect to various topologies."]},{"key":"dc:title","label":"Title","values":["Separability and metrisability in locally convex spaces"]}]}],"canonical_facts":{"dc:contributor.advisor":["Webb, John H"],"dc:creator":["Robertson, Neill Raymond Charles"],"dc:date.accessioned":["2016-09-28T18:59:39Z"],"dc:date.available":["2016-09-28T18:59:39Z"],"dc:date.issued":["1991"],"dc:description":["Bibliography: pages 58-61."],"dc:description.abstract":["This thesis is devoted to a study of the relationship between separability and metrisability in the context of locally convex spaces. The duality between sep- arability and weak*-metrisability does not carry over to non-metrisable locally convex spaces; the best that can be said in this case is that the equicontinuous subsets in the dual of a separable locally convex space are weak*-metrisable. To get around this difficulty, we often prefer to use the idea of separability by seminorm: a locally convex space E is separable by seminorm if and only if the equicontinuous subsets of its dual are weak*-metrisable. On any locally convex space E there is a finest topology Tχ which is coarser than the given topology and which makes E separable by seminorm. A question that arises is under what conditions a space E is Tχ-complete. In trying to answer this question, we are led to an intriguing binary relation which G.A. Edgar originally defined on the class of Banach spaces. In the first two Chapters of this thesis, we show that many of the results in Edgar's paper can be expressed in terms of the completeness of a space with respect to various topologies."],"dc:identifier.uri":["http://hdl.handle.net/11427/21966"],"dc:language.iso":["eng"],"dc:publisher.department":["Department of Mathematics and Applied Mathematics"],"dc:publisher.institution":["University of Cape Town"],"dc:title":["Separability and metrisability in locally convex spaces"],"dc:type":["Doctoral Thesis"],"dc:type.qualificationlevel":["Doctoral"],"dc:type.qualificationname":["PhD"]},"updated_at":"2026-07-22T22:23:09Z"}