Thin and thick sets in locally convex spaces

We introduce in this work some normed space notions such as norming, thin and thick sets in general locally convex spaces. We also study some effects of thick sets on the uniform boundedness-like principles in locally convex spaces such as “weak*-bounded sets are strong*-bounded if and only if the space is a Banach–Mackey space”. It is proved that these principles occur under some weaker conditions by means of thick sets. Further, we show that the thickness is a duality invariant, that is, all compatible topologies for some locally convex space have the same thick sets.