# Can we reduce infinitely linearly dependent spanning set to an independent set while keeping spanning?

I'm quite unfamilar to infinite-dimensional case. Let $V$ be an infinite-dimensional vector space, and let $S$ be a subset of $V$. If $S$ spans $V$ and $S$ is linearly dependent, can we know that $S$ can be reduced to a linearly independent spanning set? If so, how to prove it? Does it require AC?

• Probably a maximal linearly independent subset of $S$ will still span $V$. – Daniel Schepler Sep 15 '17 at 17:00