If not distinguished, is Cp(X) even close?

Abstract

Cp (X) is distinguished ⇔ the strong dual Lβ (X) is barrelled ⇔ the strong bidual M (X) = RX. So one may judge how nearly distinguished Cp (X) is by how nearly barrelled Lβ (X) is, and also by how near the dense subspace M (X) is to the Baire space RX. Being Baire-like, M (X) is always fairly close to RX in that sense. But if Cp (X) is not distinguished, we show the codimension of M (X) is uncountable, i.e., M (X) is algebraically far from RX, andmoreover, Lβ (X) is very far from barrelled, not even primitive. Thus we profile weak barrelledness for Lβ (X) and M (X) spaces. At the same time, we characterize those Tychonoff spaces X for which Cp (X) is distinguished, solving the original problem from our series of papers.