Johann Boos¹ and Toivo Leiger²

Copyright © 2001 Johann Boos and Toivo Leiger. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The paper aims to develop for sequence spaces E a general concept for reconciling certain results, for example inclusion theorems, concerning generalizations of the Köthe-Toeplitz duals E×(×∈{α,β}) combined with dualities (E,G),G⊂E×, and the SAK-property (weak sectional convergence). Taking Eβ:={(yk)∈ω:=𝕜ℕ|(ykxk)∈cs}=:Ecs, where cs denotes the set of all summable sequences, as a starting point, then we get a general substitute of Ecs by replacing cs by any locally convex sequence space S with sum s∈S′ (in particular, a sum space) as defined by Ruckle (1970). This idea provides a dual pair (E,ES) of sequence spaces and gives rise for a generalization of the solid topology and for the investigation of the continuity of quasi-matrix maps relative to topologies of the duality (E,Eβ). That research is the basis for general versions of three types of inclusion theorems: two of them are originally due to Bennett and Kalton (1973) and generalized by the authors (see Boos and Leiger (1993 and 1997)), and the third was done by Große-Erdmann (1992). Finally, the generalizations, carried out in this paper, are justified by four applications with results around different kinds of Köthe-Toeplitz duals and related section properties.