We work in set theory without the Axiom of Choice ZF. We prove that the Principle of Dependent Choices (DC) implies that the closed unit ball of a uniformly convex Banach space is weakly compact and, in particular, that the closed unit ball of a Hilbert space is weakly compact. These statements are not provable in ZF and the latter statement does not imply DC. Furthermore, DC does not imply that the closed unit ball of a reflexive space is weakly compact.
"Dependent Choices and Weak Compactness." Notre Dame J. Formal Logic 40 (4) 568 - 573, Fall 1999. https://doi.org/10.1305/ndjfl/1012429720