Countability Properties of Weakly Compact Sets in Asymmetric Locally Convex Spaces

Abstract

A dual pair formulation for asymmetric locally convex spaces is developed that strictly generalises the ordinary vector space setting. The concept of a polar topology carries over to the asymmetric case and some familiar results are reproduced or generalised such as the bipolar theorem and a Mackey-Arens type theorem. The implications of weak compactness and countability properties are studied and appear intimately connected to separation properties. An asymmetric analogue $C_a(K)$ of the well-known space $C_p(K)$ is introduced and the properties of (relatively) (countably respectively sequentially) compact subspaces are investigated. In particular, it is shown that $C_a([0,1])$ is not angelic. However, for Hausdorff subspaces satisfying a simple closure condition, the different compactness conditions are equivalent and imply the Fr\'echet-Urysohn property. Moreover, an analogue of the Eberlein-\v{S}mulian theorem holds in asymmetrically normed spaces.