I wonder if the span of a countable set is always a separable subspace with the norm topology? It feels like a trivial question but how does one show this?
Namely, I start of with a countable set $\{ v_n \}_{n}$ and take the span of $\{ v_n \}_n$, name it the space $A$. Take $y \in A$, then $y = \sum_{i = 1} ^N c_i v_i$ for some $N \in \mathbb{N}$. How do we show that $y \in \overline{\{ v_n \}_n}$ closure?