Denote by DC the axiom of Dependent Choices. We show that in ZF+DC, the closed unit ball of a uniformly convex Banach space is weakly compact. In particular, in ZF+DC, the closed unit ball of a Hilbert space is weakly compact (but this statement does not follow from ZF). We also show that the weak compactness of the closed unit ball of a (simply) reflexive banach space does not follow from DC.