in λ-sums of normed spaces

Abstract

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.