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## Closure and span

Suppose that $X$ is a normed vector space and $S$ is a subset of $X$. Prove that $\overline{span(S)}=\overline{span(\overline{S})}$.

Proof We have that $S\subset span(S)$. Then obviously $\overline{S}\subset \overline{span(S)}$. Since the closure of a subspace is also a subspace we have also $span(\overline{S})\subset \overline{span(S)}$. Taking the closure in this last inclusion we get
$\overline{span(\overline{S})}\subset \overline{span(S)}$.
On the other hand, we obviously have $S \subset \overline{S}$, which implies $span(S)\subset span(\overline{S})$, which implies $\overline{span(S)}\subset \overline{span(\overline{S})}$. This proves the desired equality.