In this note we introduce the concepts of Λ-Mackey sequence, Λ-Mackey convergence property, Λ-Schwartz family and associated Λ-Schwartz family and consider some applications of these to locally convex spaces. Hereby Λ denotes a Banach sequence space with the AK-property — the results of this paper generalize those in [4] where the case Λ = I1 is considered. We obtain a dual characterization of those locally convex spaces which satisfy the Λ-Mackey convergence property and characterize the dual Λ-Schwartz spaces in terms of the SM-property which is introduced in [10]. Finally, necessary and sufficient condition for a locally convex space to be ultra-bornological is proved.
A characterization of Pontryagin–van Kampen duality for locally convex spaces
