Unit ball of $X^{**}$ is weakly compact!

Question

Is it true that the closed unit ball in $X^{**}$ is compact with respect to the weak topology on $X^{**}$, where $X$ is a Banach space? If so, how can we prove it?

Answer 1

The unit ball of any Banach space $X$ is compact with respect to the weak topology if and only if $X$ is reflexive (a good exercise, which I recommend trying). Since a Banach space is reflexive if and only if $X^*$ is reflexive, we have

If $X$ is a Banach space, then the unit ball of $X^{**}$ is weakly compact if and only if $X$ is reflexive.