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Let $F$ be a metrizable locally convex topological vector space and let $F^{*}$ be its dual space endowed with the strong topology = topology of uniform convergence on (closed convex balanced) bounded subsets of $F$. Let $E\subset F^{*}$.

Is it true that $\overline{E}$ is the set of all linear functionals on $F$, which are continuous in the weak $\sigma(F,E)$ topology on every (closed convex balanced) bounded subset of $F$?

I tried to adapt the Grothendieck completion theorem to prove this, but $\sigma(F,E)$ closure of a bounded set can fail to be bounded.

If correct, I hope this result is contained in some textbook on locally convex spaces, and so a reference would be appreciated.

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