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I posted the following question in a comment at http://mathoverflow.net/questions/43986/are-there-non-reflexive-vector-spaces-isomorphic-to-their-bi-dual and it got one upvote, but it didn't get an answer, so I'll post it as an independent question.

Is there an example of an abelian Hausdorff topological group G such that G and its second dual G^^ are isomorphic as topological groups but the natural map G ---> G^^ is not a topological group isomorphism? The dual group of an abelian Hausdorff topological group is given the compact-open topology, which makes the dual group an abelian topological group, although  a priori it is not clear that G^ separates points in G, so the Hausdorfness of G^^ is part of the conditions that would need to be checked in an example (rather than being automatic).

[edit: Since G^ need not be Hausdorff, maybe I'm even willing to drop that condition. If G is an abelian top. group then G^ with the compact-open topology is an abelian top. group and G^^ is as well. Is there such G isomorphic to G^^ but not by the natural map? If a non-Hausdorff example turns out to be silly then maybe I'll stick the Hausdorff condition back in.]

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I posted the following question in a comment at http://mathoverflow.net/questions/43986/are-there-non-reflexive-vector-spaces-isomorphic-to-their-bi-dual and it got one upvote, but it didn't get an answer, so I'll post it as an independent question.

Is there an example of an abelian Hausdorff topological group G such that G and its second dual G^^ are isomorphic as topological groups but the natural map G ---> G^^ is not a topological group isomorphism? The dual group of an abelian Hausdorff topological group is given the compact-open topology, which makes the dual group an abelian Hausdorff topological group, although  a priori it is not clear that G^ separates points in G, so the Hausdorfness of G^^ is part of the conditions that would need to be checked in an example (rather than being automatic).

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# Are there non-reflexive abelian topological groups isomorphic to their second dual?

I posted the following question in a comment at http://mathoverflow.net/questions/43986/are-there-non-reflexive-vector-spaces-isomorphic-to-their-bi-dual and it got one upvote, but it didn't get an answer, so I'll post it as an independent question.

Is there an example of an abelian Hausdorff topological group G such that G and its second dual G^^ are isomorphic as topological groups but the natural map G ---> G^^ is not a topological group isomorphism? The dual group of an abelian Hausdorff topological group is given the compact-open topology, which makes the dual group an abelian Hausdorff topological group.