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 Eβ:={(yk...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2001-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171201011772 |
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| Summary: | 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. |
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| ISSN: | 0161-1712 1687-0425 |