Browsing by Author "Robertson, Neill Raymond Charles"

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    Generalized DF spaces
    (1984) Robertson, Neill Raymond Charles; Webb, John H
    A DF space is a topological vector space sharing certain essential properties with the strong duals of Frechet spaces. The class of DF spaces includes not only all such duals, but also every· normed space and many other spaces besides. The definition of a DF space is due to Grothendieck, who derived almost all the important results concerning such spaces. The "generalized DF spaces" in the title of this thesis are locally convex topological vector spaces whose topologies are determined by their restrictions to an absorbent sequence of bounded sets. In the case when this sequence is a fundamental sequence of bounded sets, we obtain the gDF spaces. Many. of the properties of DF spaces are shared by all gDF spaces.
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    Separability and metrisability in locally convex spaces
    (1991) Robertson, Neill Raymond Charles; Webb, John H
    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.