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. (shrink)

A poset is well‐partially ordered (WPO) if all its linear extensions are well orders; the supremum of ordered types of these linear extensions is the length, of p. We prove that if the vertex set X is infinite, of cardinality κ, and the ordering ⩽ is the intersection of finitely many well partial orderings of X,, then, letting, with, denote the euclidian division by κ (seen as an initial ordinal) of the length of each corresponding poset: where denotes the least (...) initial ordinal greater than the ordinal. This inequality is optimal. This result answers questions of Forster. (shrink)