In this book Perturbative Algebraic Quantum Field Theory at page 26 the author says
We equip $E'$ with the finest locally convex topology $\gamma$ that coincides with the weak one on equicontinuous sets
Where $E$ is a locally convex topological vector spaces and $E'$ its dual.
What is this topology $\gamma$?
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.