Let $H$ be a separable Hilbert space and $\{v_n\}$ be some countable subset of $H$ such that its linear span is dense in $H$. $\{v_n\}$ need NOT be linearly independent.
(Here, linear independence of a countable set means that its any finite subset is linearly independent in the sense of linear algebra.)
Then, I guess I can extract a subset of $\{v_n\}$ that is linearly independent and span the same dense subspace of $H$. But, in this process, am I required to use the Axiom of Choice?