Orthogonal complement of S and related question

Question

Let V be a real vector space with a topology induced from an inner product. Let S be a subset. Can we say either of the following ?

1. closure(S) = perp(perp(S)) , [where perp(X) = orthogonal complement of X]
2. closure(span(S)) = perp(perp(S))

Answer 1

1. is false: the orthogonal complement $S^\perp=\{x\mid\forall s\in S: x\perp s\}$ of any set $S$ is always a closed subspace, whereas the closure need not be a subspace.
2. is true: prove $S^\perp =\overline{\mathrm{span}(S)}^\perp$, and apply $U^{\perp\perp} =U$ for a closed subspace $U$.