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When saying two topological vector spaces $E$ and $F$ are in duality, does it mean that they are each other's continuous dual, i.e. $E = F^*$ and $F=E^*$, or just that one is the other's continuous dual, not necessarily true for the reverse?

If it is the former, when is $E = F^*$ and $F=E^*$ true?

For example,

a biorthogonal system is a pair of topological vector spaces $E$ and $F$ that are in duality, with a pair of indexed subsets $ \tilde v_i$ in $E$ and $\tilde u_i$ in $F$ such that $$ \langle\tilde v_i , \tilde u_j\rangle = \delta_{i,j} $$ with the Kronecker delta

Thanks and regards!

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    Yes, each is the dual of the other. But dual in the weak topology defined by the other. Which is automatically true as long as you have a bilinear pairing so that each space separates the points of the other. You can forget that last condition if you don't mind non-Hausdorff topologies.2012-03-31
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    @GEdgar: +1 Thanks as always! (1) Do you mean that two TVSes are continuous dual of each other wrt their weak topologies, if there exists "a bilinear pairing so that each space separates the points of the other"? (2) What does "a bilinear pairing so that each space separates the points of the other" mean? (3) I just found a Wiki article about [two vector spaces to be in dual pair wrt a bilinear form](http://en.wikipedia.org/wiki/Dual_pair). I wonder in your comment, if you meant two TVSes are continuous dual wrt the evaluation, where the evaluation is the bilinear form?2012-04-01

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