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Abelian groups which satisfy Pontryagin duality need not respect compactness


Authors: Dieter Remus and F. Javier Trigos-Arrieta
Journal: Proc. Amer. Math. Soc. 117 (1993), 1195-1200
MSC: Primary 22D35; Secondary 46A99
DOI: https://doi.org/10.1090/S0002-9939-1993-1132422-4
MathSciNet review: 1132422
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Abstract: Let $ {\mathbf{G}}$ be a topological Abelian group with character group $ {{\mathbf{G}}^ \wedge }$. We will say that $ {\mathbf{G}}$ respects compactness if its original topology and the weakest topology that makes each element of $ {{\mathbf{G}}^ \wedge }$ continuous produce the same compact subspaces. We show the existence of groups which satisfy Pontryagin duality and do not respect compactness, thus furnishing counterexamples to a result published by Venkataraman in 1975. Our counterexamples will be the additive groups of all reflexive infinite-dimensional real Banach spaces. In order to do so, we first characterize those locally convex reflexive real spaces whose additive groups respect compactness. They are exactly the Montel spaces. Finally, we study the class of those groups that satisfy Pontryagin duality and respect compactness.


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DOI: https://doi.org/10.1090/S0002-9939-1993-1132422-4
Article copyright: © Copyright 1993 American Mathematical Society

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