The Strict Topology and Spaces with Mixed Topologies

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Abstract
In a paper about twenty years ago, and later papers, R. C. Buck introduced a new topology, the strict topology, on spaces of continuous functions on locally compact spaces. Since then, a considerable amount of work has been done on these and similar topologies by, among others, Conway, Collins, Rubel and Shields (see references [2], [3], [4], [6], [7], [15]). In the early nineteen-fifties, the Polish mathematicians, Alexiewicz and Semadeni, considered a vector space E, on which two norms are defined, and defined a notion of convergence of sequences on E which in some sense mixed the topologies given by the norms ([1] and later papers). In 1957, Wiweger [17] showed that under natural restrictions, the space E could be given a locally convex space structure where convergent sequences were precisely the sequences considered by Alexiewicz and Semadeni. Since then, this method of mixing topologies has been studied and generalised by several mathematicians ([18], [9]).

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