Example of a subspace S of a Hilbert space such that S^(⊥⊥) does not equal S?

Question

I try to find an example of a subspace S of a Hilbert space H such that S^(⊥⊥) does not equal S. I know that subspace cannot be closed as for closed subspaces S^(⊥⊥)=S holds true.

Does there exist such a subspace? What would be a specific example or a way to construct one?

Answer 1

For an explicit example take $H=\ell^2$ and $S=\operatorname{span}\{(x_n):x_i = 1,\ x_j=0,\ i\neq j\}$. As is well known, $\bar{S} = H$, and since $(1/n)\notin S$, but $(1/n)\in H$, $S\neq H = \bar{S}$, i.e. $S$ cannot be closed. Now use the fact that $(S^\bot)^\bot = \bar{S}$.