| Summary: | A topological abelian group <i>G</i> is said to have the quasi-convex compactness property (briefly, qcp) if the quasi-convex hull of every compact subset of <i>G</i> is again compact. In this paper we prove that there exist locally quasi-convex metrizable complete groups <i>G</i> which endowed with the weak topology associated to their character groups <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mi>G</mi><mo>∧</mo></msup></semantics></math></inline-formula>, 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.
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