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) of continuous real-valued functions over a Tychonoff space X is a peculiar (although applicable) property. It is known that C_p(X) is distinguished if and only if C_p(X) is large in mathbb {R}^X if and only if X is a Delta -space (in sense of Reed) if and only if the strong dual of 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) with the compact-open topology. We also put together several known facts (Theorem 6) about distinguished spaces C_p(X) with self-contained full proofs.