Q:

Orthogonal decomposition of a Hilbert space

Let $H$ be a Hilbert space. Then, if $M$ is a closed subspace, then I know that $H= M + M^{\star}$, where $M^{\star}$ is the orthogonal complement.
If $M$ is not closed, however, I can consider its closure $\overline M $ and its orthogonal complement $\overline M ^{\star}$ so that $H = \overline M + \overline M ^{\star}$.
But,  since orthogonal complements are always closed it seems to me that $\overline M ^{\star} = M ^{\star}$. But, is it so?

A:

Since $M \subset \overline{M}$ we get $(\overline{M})^* \subset M^*$.
Next, let $x \in M^*$ and let $y \in \overline{M}$. Pick some $y_n \in M$ such that $y_n \to y$.
Then 
$$0=<x_n,y> \to <x,y>$$
Therefore 
$$<x,y>=0$$
This shows that $x \in (\overline{M})^*$. Therefore $M^*  \subset (\overline{M})^*$.

