Duality properties of spaces of non-Archimedean valued functions

Abstract

Let C(X, F) be the space of all continuous functions from the ultraregular compact Hausdorff space X into the separated locally Jf-convex space F; AT is a complete, but not necessarily spherically complete, non-Archimedean valued field and C(X, F) is provided with the topology of uniform convergence on X. We prove that C( X, F) is ^-barrelled (respectively AT-quasibarrelled) if and only if F is AT-barrelled (respectively /f-quasibarrelled). This is not true in the case of R or C-valued functions. No complete characterization of the if-bornological spaces C( X, F) is obtained, but our results are, nevertheless, slightly better than the Archimedean ones. Finally, we introduce a notion of A"-ultrabornological spaces for K non-spherically complete and use it to study Af-ultrabornological spaces C(X, F).1980 Mathematics subject classification (Amer. Math. Soc.): 46 P 05.