Orthogonality of (M)perp and (N)perp, for two linear subsets of Hilbert Space H

Question

I try to find a counterexample that, for two linear subsets $M$ and $N$ of a Hilbert space H, in general $M^⊥$ may not be orthogonal to $N^⊥$, even if $M$ ⊥ $N$. I know that if $M$ ⊥ $N$, then $(M^⊥)^⊥$ $ ⊥ $ $(N^⊥)^⊥$ but what can be said about the orthogonality of $M^⊥$ and $N^⊥$ ?

Answer 1

In $\mathbb R^3$ consider $M=\{(x,0,0):x\in \mathbb R\}$ and $N=\{(0,y,y):y\in \mathbb R\}$. Clearly they are orthogonal, but $M^\perp=\{(0,y,z):y,z\in \mathbb R\}$, and $N^\perp=\{(x,y,-y):x,y\in \mathbb R\}$. As you can verify yourself $M^\perp$ and $N^\perp$ are not orthogonal.