Dual pairs of sequence spaces - Johann Boos; Toivo Leiger (International Journal of Mathematics and Mathematical Sciences)
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.