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

Is it true that $\overline{F}=F_1$, where $F_1$ is the set of all linear functionals on $E$, whose restrictions on every (closed convex balanced) bounded subset of $E$ are continuous in the weak $\sigma(E,F)$ topology?

It seems that I can adapt the proof of Grothendieck's completion theorem to prove this. Indeed, every linear functional continuous on a closed set $B$ in $E$ can be uniformly approximated on $B$ by an element of $E^{*}$ (I guess this is Grothendieck's lemma). Hence, we only need to show that $F_1\subset E^{*}$. But this follows from the fact, that since $E$ is metrizable, $E^{*}$ is complete, and so it is a closed subset of $E'$ (the algebraic dual), andowed with the uniformity of uniform convergence on bounded sets.

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