# About Angles between subspaces of a Hilbert space

I´m working with the article "The angle between subspaces of a Hilbert Space" written by Frank Deutsch.

I need to proof the following lemma:

Let $\mathcal{M}$ and $\mathcal{N}$ be two closed subspaces of a Hilbert space $\mathcal{H}$. If $\mathcal{M}+\mathcal{N}$ is closed then $(\mathcal{M} \cap \mathcal{N})^{\perp}=\mathcal{M}^{\perp}+\mathcal{N}^{\perp}$.

Remember these following properties:

1. $(\mathcal{M}+\mathcal{N})^{\perp}=\mathcal{M}^{\perp}\cap \mathcal{N}^{\perp}$
2. $(\mathcal{M} \cap \mathcal{N})^{\perp}=\overline{\mathcal{M}^{\perp}+\mathcal{N}^{\perp}}$.

So, the following inclusion is well-known:

$\mathcal{M}^{\perp}+\mathcal{N}^{\perp} \subseteq \overline{\mathcal{M}^{\perp}+\mathcal{N}^{\perp}}=(\mathcal{M} \cap \mathcal{N})^{\perp}$.

I have no idea how to proof the other one and I tried a lot of ways as well.

Could anyone help me?

Actually, I would like to proof that $\mathcal{M}+\mathcal{N}$ is closed if and only if $\mathcal{M}^{\perp}+\mathcal{N}^{\perp}$ is closed which is really easy if the previous lemma is proved.

Another possibility is to proof that $c(\mathcal{M}, \mathcal{N})=c(\mathcal{M}^{\perp}, \mathcal{N}^{\perp})$ where $c(\mathcal{A}, \mathcal{B})$ is the cosine of the Friedrichs angle between the subspaces $\mathcal{A}$ and $\mathcal{B}$. This is because it is also well known that $\mathcal{M}+\mathcal{N}$ is closed if and only if $c(\mathcal{M}, \mathcal{N})<1$.

Unfortunately, I could not proof neither this equality. So, Can anyone drop me a hint?

Thank you! Best, Blas.

• Did you see Rudin Functional Analysis? – Ali Jan 4 '17 at 6:15
• No, I didn't. Which chapter you are suggesting me? Thank you! – Blas Fernández Jan 4 '17 at 20:48
• I´ve just looked it! I didn´t find anything useful at first sight! What are you referring to? Thank you in advance! – Blas Fernández Jan 4 '17 at 21:30