A generalized inductive limit strict topology on the space of bounded continuous

A generalized inductive limit strict topology beta(infinity) is defined on C-b(X, E), the space of all bounded, continuous functions from a zero-dimensional Hausdorff space X into a locally K-convex space E, where K is a field with a nontrivial and nonarchimedean valuation, for which K is a complete ultrametric space. Many properties of the topology beta(infinity) are proved and the dual Of (C-b (X, E), beta(infinity)) is studied.