# Topological vector spaces question regarding dual spaces

there's a question that I've be puzzling over for a little while now and I haven't made much progress, so I thought I'd better ask for some help. It is as follows:

Suppose that $\langle E,F\rangle$ are a pair of dual spaces. Prove that $E$ is $\sigma(E,F)$-complete iff $F$ is isomorphic to $E^*$.

I've managed to prove that $F$ isomorphic to $E^*$ implies that $E$ is $\sigma(E,F)$-complete, but I can seem to do it the other way around. Any suggestions?

Thanks Andrew

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It might be good to include a definition for "$\sigma(\cdot,\cdot)$ complete", because some people might not recognize/remember it. – rschwieb Oct 18 '12 at 15:17
@Andrew: Are you sure? I was able to prove $F$ isomophic to $E^*$, if $F$ is $\sigma(F,E)$-complete. – Vobo Oct 19 '12 at 19:17
Vobo, to be honest I think that you are correct. It certainly holds true if one states it like that. Perhaps it was a misprint. Does anyone know whether it is true? – Andrew Oct 27 '12 at 8:56