Cp(X) in coreflective classes of locally convex spaces

Situations analogous to some classical characterization are investigated, of topological spaces X for which Cp(X) belongs to a given coreflective class C of locally convex spaces. For instance, if C contains all strong Mazur spaces and is contained in the class of weak Mazur spaces, then Cp(X) belongs to C iff X is realcompact. If C is the coreflective hull of Rω and X is a P-space, then Cp(X) belongs to C iff X is realcompact.