Krein’s Theorem in the Context of Topological Abelian Groups

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