Introducing a dual space structure

Question

Suppose that we have a Banach space $X$ together with some locally convex Hausdorff topology on $X$, weaker than the one given by the norm, which makes the unit ball of $X$ compact. Is $X$ (Banach-space) isomorphic to $Y^*$ for some Banach space $Y$?

I am aware of a related result of Kajser which says that if $X$ is a Banach space and $(f_i)$ is a separating family of bounded functionals on $X$ which makes the ball of $X$ compact in the topology $\sigma(X, (f_i))$, then $X\cong \overline{\mbox{span}}\{f_i\colon i\in I\}^*$, however I am not sure if the former can be deduced out of it.

Answer 1

Yeah, this is true. Let ${\cal T}$ be an LCH topology on $X$ which makes the unit ball compact. Then every ${\cal T}$-continuous linear functional takes the unit ball to a compact subset of the scalar field and hence is bounded for the norm. We also know from general theory that the ${\cal T}$-continuous linear functionals $f_i$ separate points, and that the $\sigma(X, (f_i))$ topology is weaker than ${\cal T}$, so the conclusion now follows from the result you quoted.