Completeness in the Mackey topology

Bonet and Cascales [Non-complete Mackey topologies on Banach spaces, Bulletin of the Australian Mathematical Society, 81, 3 (2010), 409–413], answering a question of M. Kunze and W. Arendt, gave an example of a norming norm-closed subspace N of the dual of a Banach space X such that μ(X, N) is not complete,where μ(X, N) denotes the Mackey topology associated with the dual pair 〈X, N〉. We prove in this note that we can decide on the completeness or incompleteness of topologies of this form in a quite general context, thus providing large classes of counterexamples to the aforesaid question. Moreover, our examples use subspaces N of X* that contain a predual P of X (if exists), showing that the phenomenon of noncompleteness that Kunze and Arendt were looking for is not only relatively common but illustrated by “well-located” subspaces of the dual. We discuss also the situation for a typical Banach space without a predual—the space c0—and for the James space J.