Krein’s Theorem in the Context of Topological Abelian Groups

Abstract

A topological abelian group G is said to have the quasi-convex compactness property (briefly, qcp) if the quasi-convex hull of every compact subset of G is again compact. In this paper we prove that there exist locally quasi-convex metrizable complete groups G which endowed with the weak topology associated to their character groups G∧, do not have the qcp. Thus, Krein’s Theorem, a well known result in the framework of locally convex spaces, cannot be fully extended to locally quasi-convex groups. Some features of the qcp are also studied.

Keywords

• quasi-convex subset
• determining subgroup
• quasi-convex compactness property
• Krein’s Theorem