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

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?

• "Closed unit ball of $Y^*$ is compact in weak topology for any Banach space". I don't think this is true – NewB Feb 8 '19 at 15:32
• @mathworker21 This is true for the weak$^*$-topology, not the weak topology. – Aweygan Feb 8 '19 at 15:40
• my apologies you guys – mathworker21 Feb 8 '19 at 15:49

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.
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.