Completeness in the Mackey topology by norming subspaces

• Published in: Journal of mathematical analysis and applications 2019-10, Vol.478 (2), p.776-789, Article 776
• Main Authors: Guirao, A.J., Martínez-Cervantes, G., Rodríguez, J.
• Format: Article
• Language: English
• Subjects: Completeness; Mackey topology; Mazur property; Norming subspace
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2019.05.054

We study the class of Banach spaces X such that the locally convex space (X,μ(X,Y)) is complete for every norming and norm-closed subspace Y⊂X⁎, where μ(X,Y) denotes the Mackey topology on X associated to the dual pair 〈X,Y〉. Such Banach spaces are called fully Mackey complete. We show that fully Mackey completeness is implied by Efremov's property (E) and, on the other hand, it prevents the existence of subspaces isomorphic to ℓ1(ω1). This extends previous results by Guirao et al. (2017) [9] and Bonet and Cascales (2010) [3]. Further examples of Banach spaces which are not fully Mackey complete are exhibited, like C[0,ω1] and the long James space J(ω1). Finally, by assuming the Continuum Hypothesis, we construct a Banach space with w⁎-sequential dual unit ball which is not fully Mackey complete. A key role in our discussion is played by the (at least formally) smaller class of Banach spaces X such that (Y,w⁎) has the Mazur property for every norming and norm-closed subspace Y⊂X⁎.