# Thread: Infinite dimensional vector space bases.

1. ## Infinite dimensional vector space bases.

Here is a question I was wondering. It seems it should be true, but I can't figure it out.

Let $V$ be an $\mathbb{R}$-vector space and let $S \subset V$ be a nonempty subset of $V$. Let $W$ be the vector space generated by $S$. Does $S$ contain a basis for $W$?

I am aware that this is trivial if $V$ is finite-dimensional, but what about in the infinite-dimensional case? I have the feeling that it requires the axiom of choice.

2. ## Re: Infinite dimensional vector space bases.

Update: I managed to do it using Zorn's lemma, essentially following the proof at PlanetMath with a few minor changes to the proof.