Some remarks about Mackey convergence (Q1902914)

Summary: We examine Mackey convergence with respect to \(K\)-convergence and bornological (Hausdorff locally convex) spaces. In particular, we prove that: Mackey convergence and local completeness imply property \(K\); there are spaces having \(K\)-convergent sequences that are not Mackey convergent; there exists a space satisfying the Mackey convergence condition, is barrelled, but is not bornological; and if a space satisfies the Mackey convergence condition and every sequentially continuous seminorm is continuous, then the space is bornological.