Properties of the weak and weak* topologies of function spaces

Let X be a Tychonoff space, and let S be a directed family of functionally bounded subsets of X containing all finite subsets of X. Denote by CTS (X) the space of all continuous functions on X endowed with the topology of uniform convergence on the sets of the family S. We characterize X for which the space CTS (X) endowed with the weak topology satisfies numerous weak barrelledness conditions or (DF)-type properties, or it has a locally convex property stronger than the property of being a Mackey space. It is shown that the dual space of C-TS ( X) is weak* sequentially Ascoli iff X is finite. We prove also that if C-TS (X) is an l(infinity)-quasibarrelled space, then the strong dual of C-TS (X) is a weakly sequentially Ascoli space iff X is finite.