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?
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
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