On the weighted function space CVo(X,E)CVo(X,E) in the non-Archimedean setting

For VV a Nachbin family, on a zero-dimensional Hausdorff topological space XX, and EE a non-Archimedean Hausdorff locally convex space, it is shown that the dual spaces of the Nachbin spaces CVo(X)CVo(X) and CVo(X,E)CVo(X,E) are algebraically isomorphic to certain spaces of measures on a ring of subsets of XX. Also, the space of all continuous linear maps, from CVo(X,E)CVo(X,E) to another locally convex space FF, is algebraically isomorphic to a space of L(E,F)L(E,F)-valued measures.