Integral Representations of Continuous Linear Maps On p-adic Spaces of Continuous Functions

Let X be a Hausdorff zero-dimensional topological space, K (X) the algebra of all clopen subsets of X, E and F Hausdorff locally convex spaces over a non-Archimedean valued field K, C-b(X, B) the space of all bounded continuous E-valued functions on X and L(E, F) the space of all continuous linear operators from E to F. The space M(K (X), L(E, F)), of all finitely-additive L(E, F)-valued measures m on K (X) for which m(K (X)) is an equicontinuous subset of L(E, F), is investigated. If we equip C-b(X, E) with the topologies beta(o), beta, beta(u) or beta(e) and if F is complete, it is shown that, the corresponding spaces of all continuous linear operators from C-b(X, E) to F are algebraically isomorphic to certain subspaces of M (K (X), L(E, F)).