Barreled topology finer than the weak-star topology

Question

I have an infinite-dimensional (Hausdorff separated, non-metrizable) locally convex space $(X,\tau)$ with topological dual $X^*$.

My questions are:

Under what conditions is there a barreled topology on $X^*$ that is finer than the weak-star topology?

If $(X,\tau)$ is complete then is the answer to the previous question positive?

Answer 1

Hint: For any vector space $Y$, the finest (the biggest) topology that makes $Y$ into a locally convex topological vector space is $\sigma(Y,Y') $ Where $Y'$ is the set of all linear functionals on $Y$! This topology is called core convex topology and indeed it is a barreled topology, You can verify this easily via looking at the Topological basis of this topology look at my paper in arXiv https://128.84.21.199/abs/1704.06932v1 .

So Now Put $Y = X^*$