# Orthogonal complement of S and related 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))

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 $$U$$.
• Need not be a subspace. For example, a closed ball is closed but not a subspace. Its perp is $\{0\}$. Dec 7, 2019 at 15:00