Pointwise bornological vector spaces

The existing duality between topological and bornological vector spaces allows us to define bornological objects in the category of topological vector spaces. For a Tychonoff space X and a set B of relatively pseudocompact subsets of X, the vector space C(X) endowed with the topology of uniform convergence on elements of B is a locally convex topological vector space, the bornological coreflection of which is described in [J. Schmets, Espaces de Fonctions Continues, Lecture Notes in Math., vol. 519, Springer-Verlag, Berlin, Heidelberg, New York, 1976; J. Schmets, Spaces of Vector-Valued Functions, Lecture Notes in Math., vol. 1003, Springer-Verlag, Berlin, Heidelberg, New York, 1983; J. Dontchev, S. Salbany, V. Valov, Barrelled and bornological function spaces, J. Math. Anal. Appl. 242 (2000) 1–17; J. Schmets, Spaces of vector-valued continuous functions, in: Proceedings Vector Space Measures and Applications I, Dublin, 1977, in: Lecture Notes in Math., vol. 644, Springer-Verlag, Berlin, Heidelberg, New York, 1978, pp. 368–377; J. Schmets, Bornological and ultrabornological C(X,E) spaces, Manuscripta Math. 21 (1977) 117–133]. If the elements of B are not supposed to be relatively pseudocompact, then this topology is no longer a vector topology and the bounded sets do not form a bornology, so the classical theory on bornologicity cannot be applied to it. The aim of this paper is to extend the duality between topological and bornological vector spaces to larger classes of objects and, moreover, apply it to C(X) endowed with three different, natural topologies.