The strict topology and spaces with mixed topologies

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]). The purpose of this note is to show that the strict topology for function spaces is a special case of a mixed topology. We then intend to use the theory of mixed topologies to give quick proofs of the basic results on strict topologies. This has the advantage of giving simpler proofs and of eliminating some heavy analysis. It also allows the definition of strict topologies in a more general setting than has been considered until now and in some cases gives new results for the standard situation.