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 $\mu(X,N)$ is not complete, where $\mu(X,N)$ denotes the Mackey topology associated with the dual pair $\langle X,N\rangle$. 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 $c_0$—and for the James space $J$.