THE DUAL OF THE LOCALLY CONVEX SPACE C-p (X)

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.