Can any spanning subset be reduced to a Hamel basis in infinite dimensions?

Question

As per this question, it is possible to extend any l.i. subset of vectors $A \subset V$ of a vector space to a (Hamel) basis, $B \subset V$. Is it likewise possible to delete vectors from a spanning subset $S\subset V$ ($span(S) = V$)? The argument is trivial in finite-dimensions, but it's not clear to me that this is possible in infinite dimensions. Even a sketch of the proof would be greatly appreciated.

Answer 1

Let $B$ be a maximal independent subset of $S$, which exists by Zorn’s Lemma. Then we can show that $span(B) = span(S) = V$. For given $s \in S$, suppose $s \notin span(B)$. Then $B \cup \{s\}$ is a larger independent subset of $S$ than $B$ is; contradiction.

Also, note that we cannot prove this theorem without Zorn’s Lemma (or some statement of the equivalent axiom of choice). In fact, over ZF, the claim that every vector space has a basis is equivalent to Zorn’s Lemma. Clearly, your claim implies that all vector spaces have bases, as we can take $S = V$.