I am looking for an example of a semi-reflexive locally convex topological vector space, whose strong dual is not semi-reflexive. Is there some well-known example ?
2 Answers
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The standard example is the dual of a non reflexive Banach space under the Mackey topology from its predual. See e.g. Porblem 20.A of Kelley & Namioka's "Linear Topological Spaces". This book has a pretty good discussion of semi-reflexivity.
answered Jan 29, 2014 at 18:49
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$\begingroup$ Whether you take the Mackey or the weak*-topology does not really matter. So, this is essentially the same example as in my answer. $\endgroup$ Commented Jan 31, 2014 at 12:14
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$\begingroup$ Right, Jochen; I should have noted that. I wanted to give a book reference. $\endgroup$ Commented Feb 1, 2014 at 16:35
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I think that semi-reflexivity means that the space equals its bidual algebraically (but not necessarily topologically).
Take a non-reflexive Banach space $X$ (like $X=c_0$) and endow the dual $Y=X'$ with the weak* topology. Then $Y'=X$ so that $Y''=Y$ but the strong dual $X$ is not reflexive.
answered Jan 29, 2014 at 7:28
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$\begingroup$ I meant $Y$ to be the counterexample. $\endgroup$ Commented Jan 31, 2014 at 12:12