in λ-sums of normed spaces

Let E n , n=1,2,··. be a sequence of normed spaces and λ an AK-Köthe sequence space. The λ-sum of the spaces E n is defined by λ{E n }:={(x n ) n : x n is in E n and (|x n |) n ∈λ}. Starting from the topology in λ, λ{E n } can be given a natural topology. In this paper we characterize the dual of λ{E n } as the λ × -sum of E n ' and give conditions for λ{E n } to be quasi-barrelled, barrelled, reflexive, bornological or distinguished

Köthe echelon spaces à la Dieudonné

Let (gn) be a sequence of locally integrable functions defined on a Radon measure space. The echelon space associated to (gn) was defined by J. Dieudonné as the Köthe-dual of (gn), i.e. the space Λ of all locally integrable functions f such that all the integrals ∫ |f·gn| are finite. Denote by Λx the Köthe-dual of Λ. We prove that Λ(β(Λ,Λx)) is a Fréchet space with dual Λx. This result gives its correct sense to a wrong affirmation of J. Dieudonné and validates those instances where it has been used. As a tool to prove this result, we study the problem of when the strong dual of a perfect space coincides with its Köthe-dual and give some necessary and sufficient conditions

Duals of vector-valued Köthe function spaces

Let Λ be a perfect Köthe function space in the sense of Dieudonné, and Λ× its Köthe-dual. Let E be a normed space. Then the topological dual of the space Λ(E) of Λ-Bochner integrable functions equals the corresponding Λ×(E′) if and only if E′ has the Radon–Nikodým property. We also give some results concerning barrelledness for spaces of this kind

A note on the Köthe dual of Banach-valued echelon spaces

Several different ways of defining the Köthe dual of echelon spaces of Banach-valued functions are shown to be equivalen