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.

Answer 2

It is not true in general. The unit ball in an arbitrary Banach space is weakly compact if and only this Banach space is reflexive (see here for references and a proof sketch). The second dual of a Banach space is not necessarily reflexive; in fact, the dual of a Banach space is reflexive if and only if the Banach space itself is reflexive (see here). Thus, the unit ball in $Y^{\ast\ast}$ is weakly compact if and only if $Y$ is reflexive.