A universal topology for sequence spaces

Abstract

In this paper we introduce a locally convex topology which is defined on every space S of sequences, namely, the inductive topology determined by all locally convex FK-spaces which are contained in S. It is shown that every matrix mapping between such spaces is continuous. The main result is Theorem 3.2 which relates the extent of the multiplier algebra M(S) of a sequence space S with some of its topological properties in this topology. Many attempts have been made to find a "natural" topology for an arbitrary space of sequences. The first was that of K6the and Toeplitz [8] in 1934 which introduced the notion of duality. This concept was generalized or refined in the following works among others: [2-4, 6, 9, 12]. An entirely different idea for topologizing sequence spaces is that of FK-space introduced by Zeller (see [-17]). Zeller also classified FK-spaces according to their sectional convergence properties [t7, 18]. FK-spaces have proven useful in function theory and summability, but many spaces, in particular dual spaces of non-Banach FK-spaces have no FK-topology. Two other ideas for topologizing sequence spaces were offered in [12]. One was a broad generalization of the K6the-Toeplitz theory and the other employed the natural duality between an arbitrary sequence space and the space ~o of finitely non-zero sequences. In Section 2 we shall define the IK-topology, a direct generalization of FK-topology which applies to all sequence spaces. We then prove that matrix mappings between spaces having such topologies are continuous. This extends a desirable property of FK-spaces.