Suppose $E$ is a Banach space, and $K\subseteq E^*$ is convex, and is closed and bounded with respect to weak-* topology. Is it true that $K$ is compact?
If $E$ is reflexive, then this is the case, since weak boundedness implies norm boundedness, and it easily follows from Banach-Alaoglu theorem that $K$ is compact (even without convexity). I wonder what happens if we drop the reflexive condition.
Any ideas? Thanks!
As user1952009 pointed out, the preceding argument also works for non-reflexive Banach spaces, see Given a Banach space $X$, are weak$^*$ bounded subsets of the dual space $X '$ also strongly bounded (with respect to the usual norm in $X '$)?.