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 CTS(X) is weak^∗ sequentially Ascoli iff X is finite. We prove also that if CTS(X) is an ℓ_∞-quasibarrelled space, then the strong dual of CTS(X) is a weakly sequentially Ascoli space iff X is finite.