In a normed space $X$ is there an equivalence between these two proposition?
$1)$ $X$ is reflexive;
$2)$ $B$, the unit ball of $X$, is weakly compact.
A proof of this theorem can be found in:
See Theorem 3.31.
Edit: The referenced theorem assumes that $X$ is Banach; however, this automatically follows from either of conditions (1) and (2):