Feral dual spaces and (strongly) distinguished spaces C(X)

Abstract

Following Dieudonné and Schwartz a locally convex space is distinguished if its strong dual is barrelled. The distinguished property for spaces $$C_p(X)$$ C p ( X ) of continuous real-valued functions over a Tychonoff space X is a peculiar (although applicable) property. It is known that $$C_p(X)$$ C p ( X ) is distinguished if and only if $$C_p(X)$$ C p ( X ) is large in $$\mathbb {R}^X$$ R X if and only if X is a $$\Delta $$ Δ -space (in sense of Reed) if and only if the strong dual of $$C_p(X)$$ C p ( X ) carries the finest locally convex topology. Our main results about spaces whose strong dual has only finite-dimensional bounded sets (see Theorems 2, 7 and Proposition 4) are used to study distinguished spaces $$C_k(X)$$ C k ( X ) with the compact-open topology. We also put together several known facts (Theorem 6) about distinguished spaces $$C_p(X)$$ C p ( X ) with self-contained full proofs.