What is the finest locally convex topology that coincides with the weak one on equicontinuous sets

Solution 1:

As far as I remember, it is a theorem of Grothendieck that this locally convex topology is precisely the topology of unifom convergence on the pre-compact subsets of $E$ which is thus generated by the seminorms $$p_K(\varphi)=\sup\{|\varphi(x)|: x\in K\}$$ with all such sets $K$. This should be in utmost generality in Koethe's book §21.7.

For complete locally convex spaces $E$, a version of the theorem is in the book Introduction to Functional Analysis of Meise and Vogt, lemma 24.21.

Edit. The finest locally convex topology which agrees with the weak$^*$-topology on all equicontinuous sets is the topology of uniform convergence on pre-compact sets only for metrizable locally convex spaces. This is (a version of) the Banach-Dieudonné theorem.