On the topology of simple convergence in non-Archimedean function spaces (Q1077681)

Let X be an ultraregular Hausdorff space, F a non-trivial complete non- Archimedean valued field and E a non-Archimedean F-locally convex space over F. Let \(C_ s(X,E)\) be the space of all continuous E-valued functions on X with the topology of simple convergence. Among the results we mention that for E a normed space: (1) \(C_ s(X,E)\) is bornological iff X is \({\mathbb{N}}\)-replete. (2) A linear form \(\phi\) on \(C_ s(X,E)\) is continuous iff it is continuous on bounded sets. (3) The strong dual of \(C_ s(X,E)\) coincides with the strong dual of the subspace of \(C_ s(X,E)\) of functions with relatively compact range. Readers interested in (1) might also be interested in a forthcoming paper of the reviewer in J. Austral. Math. Soc.