Suppose that $E$ is a Banach space and let $E^*$ denote its dual space with canonical norm $\lVert\bullet\rVert_{E^*}$. Suppose that $\lvert\bullet\rvert_{E^*}$ is an equivalent norm on $E^*$. The Banach–Alaoglu theorem tells us that the unit ball with respect to the norm $\lVert\bullet\rVert_{E^*}$ is weak-* compact. Is it true that the unit ball with respect to the norm $\lvert\bullet\rvert_{E^*}$ is also weak-* compact?
More generally, if $E$ is a locally convex topological vector space, and the polar topology of $E^*$ is induced by some norm $\lvert\bullet\rvert_{E^*}$. Is it true that the unit ball is weak-* compact?
If $E$ is reflexive, then the unit ball of $E^*$ with respect to the norm $\lvert\bullet\rvert_{E^*}$ is strongly closed and convex, hence weakly closed and therefore weak-* closed. I don't know in general, whether it's true.
Any ideas? Thanks!