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

Question

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 $X=c_0$?

Answer 1

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.

Answer 2

Krein–Milman theorem — A compact convex subset of a Hausdorff locally convex topological vector space is equal to the closed convex hull of its extreme points.

Since the unit ball of $c_0$ has no extreme points, we are done.