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 $X=c_0$?
Best Answer
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.