It is known that if every vector space has a basis, then the axiom of choice holds. Is the weaker claim that every normed space (over $\mathbb{R}$ or $\mathbb{C}$) has a basis enough to prove $AC$?
I'm also interested in the stronger assumption the this is true for "normed" vector fields over arbitrary ordered fields, or the same idea with some other generalization of a norm I may not know about.
