The dual of the locally convex space $C_p(X)$

Abstract

If $X$ is an infinite Tichonov space, we show that the weak dual $L_{p}(X)$ of the continuous function space $C_{p}(X)$ cannot be barrelled, bornological, or even quasibarrelled. Indeed, of the fourteen standard weak barrelledness properties between Baire-like and primitive, $L_{p}(X)$ enjoys precisely the four between property (C) and primitive if $X$ is a P-space, and none otherwise. Since $L_{p}(X)$ is $S_{\sigma}$, it must admit an infinite-dimensional separable quotient. Under its Mackey topology, $L_{p}(X)$ enjoys eleven of the properties if $X$ is discrete, nine if $X$ is a nondiscrete P-space, and none otherwise.