Double orthogonal complement is equal to topological closure

Let $U=\operatorname{span}(A)$. Then it's easy to see that $$ U^{\perp}=A^{\perp} $$ It's also easy to see that $$ \overline{U}^\perp=U^{\perp} $$ (where $\overline{U}$ denotes the closure of $U$) using continuity of the inner product. Thus $$ \overline{U}=\overline{U}^{\perp\perp}=U^{\perp\perp} $$ assuming you know that, for a closed subspace $V$, $V=V^{\perp\perp}$.