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?
This statement is in fact stronger than the claim that every vector space has a basis, which requires the full axiom of choice. So the answer is yes, but only with choice.