Köthe echelon spaces à la Dieudonné

Abstract

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.