Question regarding direct products of closed subspaces of a Hilbert space

Question

Let $H$ be a Hilbert space, and let $M,N$ be closed subspaces of $H$, is it true that $M+N\subseteq \overline{ M \cup N} $? Or is there other necessary conditions for this to be true like $M,N$ must be orthogonal from each other?

Answer 1

No, consider the Hilbert space $\mathbb{R}^2$. Let $X$ be subspace generated by $(1,0)$ and $Y$ be the subspace generated by $(0,1)$. Now $\overline{X\cup Y}=X\cup Y$ since the finite union of closed sets is closed. But $\mathbb{R}^2=X+Y\not\subset X\cup Y$. Moreover, in this example, $X$ and $Y$ are orthogonal spaces.