Distinguished Cp(X) spaces

Abstract

We continue our initial study of Cp(X) spaces that are distinguished, equiv., are large subspaces of RX , equiv., whose strong duals Lβ(X) carry the strongest locally convex topology. Many are distinguished, many are not. All Lβ(X) spaces are, as are all metrizable Cp(X) and Ck (X) spaces. To prove a space Cp(X) is not distinguished, we typically compare the character of Lβ(X) with |X|. A certain covering for X we call a scant cover is used to find distinguished Cp(X) spaces. Two of the main results are: (i) Cp(X) is distinguished if and only if its bidual E coincides with RX , and (ii) for a Corson compact space X, the space Cp(X) is distinguished if and only if X is scattered and Eberlein compact.