Separability and metrisability in locally convex spaces

• Main Author: Robertson, Neill Raymond Charles
• Other Authors: Webb, John H
• Format: Doctoral Thesis
• Language: English
• Published: University of Cape Town 2016
• Subjects: Mathematics and Applied Mathematics
• Online Access: http://hdl.handle.net/11427/21966

Bibliography: pages 58-61. === 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.