Orthogonal complements are same implies subspaces are same

Question

Let $H$ be Hilbert space and let $Y$ and $M$ be any subspaces of $H$ such that $M'=Y'$ then is it true that $M=Y$, where $'$ denotes the orthogonal complement. If one of them is closed then equality is trivial but I was wondering what happens if we remove the condition of closedness?

Answer 1

If $M=H$ and $Y$ is a dense proper subspace then the orthogonal complements are both $\{0\}$ but $M \neq Y$.

Specific example: $M=H=\ell^{2}$ and $Y=\ell_0$ the space of sequences which are eventually $0$. If you want an example where both $M$ and $Y$ are non-closed subspaces replace $M$ by $\ell^{1}$.