Remarks on spherical completeness of non-archimedean valued fields (Q1344162)

Let \(K\) be a non-Archimedean valued field. \textit{J. von Tiel} proved in [Indag. Math. 27, 249-289 (1965; Zbl 0133.065)]\ that if \(K\) is spherically complete then every locally convex space \((E, \tau)\) over \(K\) admits the Mackey topology (i.e. the finest locally convex topology \(\mu\) for which \((E, \tau)'= (E, \mu)'\)). In this note the following converse is proved. If for every local convex space over \(K\) the Mackey topology exists then \(K\) is spherically complete. The proof is elegant and ingenious. Furthermore, it is shown that each one of the related statements `the completion of a dual-separating \(K\)-normed space is separating' and `every closed subspace of a dual-separating \(K\)-normed space is weakly closed' is equivalent to spherical completeness of \(K\).