Mackey hyperplanes/enlargements for Tweddle's space
This paper investigates properties of different classes of locally convex spaces related to barrelled spaces and spaces endowed with the Mackey topology. The motivation for the present research comes from an article by \textit{I. Tweddle} [Indian J. Math. 50, No. 1, 99--114 (2008; Zbl 1157.46002)] that defines locally convex topologies in the Banach space \(\ell_\infty\) of bounded sequences that are complete and Mackey spaces (i.e., they are endowed with their Mackey topology), and satisfy that every hyperplane of the space endowed with this topology is also Mackey. Among other results, the author shows that an example due to \textit{I. Tweddle} and him [Bull. Aust. Math. Soc. 51, No. 3, 521--528 (1995; Zbl 0838.46002)] shares the same behavior. The paper mainly discusses the following question: Let \(E\) be a Mackey space such that its topological dual does not coincide with the algebraic dual. Which is the weakest barrelledness condition on \(E\) to ensure (Q1) that every dense hyperplane is Mackey, (Q2) that at least one dense hyperplane is Mackey? The related question of the existence of Mackey enlargements is also considered. Many properties of Tweddle's examples are analyzed.
scientific article; zbMATH DE number 6341690
scientific article