# Banach spaces admitting a coarser topology for which the closed unit ball is compact.

Let $$X$$ be a Banach space and let $$B$$ be its closed unit ball.

It is well known that $$B$$ is compact in the weak topology provided $$B$$ is reflexive. Otherwise, if $$X$$ is at least a dual space, then $$B$$ is compact in the weak* topology.

These examples show that, in many situations, there exists a Hausdorff locally convex topology on $$X$$, coarser than the norm topology, relative to which $$B$$ is compact.

For $$X=c_0$$, I cannot think of such a topology and I doubt it exists. The reason is I feel the sequence $$\{x_n\}_n$$ given by $$x_n= (1,1,\ldots, 1,0,0, 0\ldots)$$ ($$n$$ ones) cannot possibly have a cluster point in any sensible topology.

Question. Given a Banach space $$X$$, is there always a Hausdorff locally convex topology on $$X$$, coarser than the norm topology, relative to which the closed unit ball is compact? Can one characterize the spaces for which this is true? What if $$X=c_0$$?

It is an old result of K.F. Ng [On a theorem of Dixmier, Math. Scand. 29 (1971), 279–280 (1972)] that a Banach space $$X$$ whose unit ball $$B$$ is compact in some coarser locally convex topology $$\tau$$ is a dual Banach space. Namely, it is the dual of $$Y=\{f\in X^*: f|_B$$ is $$\tau$$-continuous$$\}$$ endowed with the dual norm.
Since the unit ball of $$c_0$$ has no extreme points, we are done.