Dual pairs of sequence spaces

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 E β : = { ( y k ) ∈ ω : = 𝕜 ℕ | ( y k x k ) ∈ c s } = : E c s , where c s denotes the set of all summable sequences, as a starting point, then we get a general substitute of E c s by replacing c s 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 , E S ) 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 , E β ) . 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.