Prove that all vector spaces (not necessarily of finite dimension) have a base.

Question

jamesj · Answer

A basis, you mean?

turingtest · Answer

yeah, sorry, I translated this from spanish...

jamesj · Answer

This is not trivial.  You're going to need the axiom of choice or one of the equivalent statements.  Do you have a proof yourself already?

turingtest · Answer

No, I have no proof prepared, nor do I know where to find one. Plus my linear algebra skills are mediocre at best. This problem is from the same entrance exam as those calculus problems we did earlier. I posted it just to see what the solution may look like, and what kind of theorems I might need to know. I strongly doubt that I have anything to contribute here.

jamesj · Answer

It's a bit of pain to write out, so I'll link to a perfectly good one: http://mathprelims.wordpress.com/2009/06/10/every-vector-space-has-a-basis/ Zorn's Lemma is an important tool in higher mathematics. It turns out it is logically equivalent to the Axiom of Choice.