Dual pairs of sequence spaces

Abstract

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 SAKproperty (weak sectional convergence). TakingE cs , where cs denotes the set of all summable sequences, as a starting point, then we get a general substitute of E cs 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, 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 Leiger (1993 and), 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.2000 Mathematics Subject Classification. 46A45, 46A20, 46A30, 40A05.

Dual pairs of sequence spaces. III

“…In the sequel, we assume that S is a fixed BK-space containing ϕ such that the BK-topology τ S of S is a ϕ-topology, and there exists a sum s ∈ S on S, that is, Other examples are given in [3,4].…”

“…In [3,4] the authors defined and studied dual pairs (E, E S ) where E is a sequence space, S is a K-space on which a sum s is defined in the sense of Ruckle [14], and E S is the space of all factor sequences, that is the set of all sequences u = (u k ) with (u k x k ) ∈ S for each x ∈ E. In that general situation, among other things, several problems related to convergence or boundedness of sections of sequences in K-spaces are investigated. In generalization of the SAK-property (weak sectional convergence) in the case of the dual pair (E, E β ), where E β = E cs (cs denotes the space of all convergent series), the SK-property was introduced and examined in [3] in the general situation of dual pairs (E, E S ).…”

“…In generalization of the SAK-property (weak sectional convergence) in the case of the dual pair (E, E β ), where E β = E cs (cs denotes the space of all convergent series), the SK-property was introduced and examined in [3] in the general situation of dual pairs (E, E S ).…”

“…In the previous paper[4] dual pairs (E, E S ) were considered where is a directed index set) and the sum s ∈ S has the representation…”

Solid and H-solid topologies on sequence groups

“…In the case of groups of sequences on a group, we will give adequate definitions for solids and H−solid parts taking into account the existence of only of internal law, and we define a solid topology and H−solid topology on a group of sequences. In [1,Theorem 6.3], Boos and Leiger extended wellknown inclusion theorems of Bennet and Kalton to a solid sequence spaces in the classical case. In non-archimedean analysis (n.a), De Grande-De Kimpe [2] gave a characterization of the natural topology N a over a perfect sequence space; in particular she demonstrated that this topology is solid; this result was generalized on a sequence space E(X) over a topological vector space X by Ameziane Hassani, El amrani and Babahmed in [3]; we gave also some results concerning polar and solid topology on E(X).…”