Compact convergence and the order bidual forC(X)

An order-theoretic characterization of the topology of compact convergence on the lattice C(X) of all continuous real-valued functions on X is provided for a realcompact space X, analogous to the order unit characterization for compact X. The approach is to generalize the concept of an order unit to permit consideration of locally convex topologies. The characterization is then achieved by viewing C(X) as a subspace of its order bidual. In addition, the bidual is employed to provide an order-theoretic description of the continuous convergence structure on C(X).