Update: I managed to do it using Zorn's lemma, essentially following the proof at PlanetMath with a few minor changes to the proof.
Here is a question I was wondering. It seems it should be true, but I can't figure it out.
Let be an -vector space and let be a nonempty subset of . Let be the vector space generated by . Does contain a basis for ?
I am aware that this is trivial if is finite-dimensional, but what about in the infinite-dimensional case? I have the feeling that it requires the axiom of choice.