Hilbert spaces and orthogonality sets

Question

I need to prove if $X$ is a Hilbert space and $M$ and $N$ it's closed: $$ (M+N)^\perp=M^\perp\cap N^\perp $$ thanks

Answer 1

This is true for every sets $M$ and $N$, in any inner-product space $X$. They don't need to be subspaces, and they don't need to be closed. Note that the orthogonal of any set is always a closed subspace.

In words: a vector $x$ is orthogonal to $M+N$ if and only if it is orthogonal to $M$ and $N$.

Sufficient: assume $(x,m)=0$ for all $m\in M$ and $(x,n)=0$ for all $n\in N$. Then $(x,m+n)=(x,m)+(x,n)=0$ for all $m\in M$ and all $n\in N$. So $(x,k)=0$ for all $k=m+n\in M+N$.

Necessary: assume $(x,k)=0$ for all $k\in M+N$. Since $M\subseteq M+N$,, this implies $(x,m)=0$ for all $m\in M$. Likewise, $(x,n)=0$ for all $n\in N$.

Now a more instructive way for the "necessary" direction: first observe the fundamental related fact $$ S\subset T\quad\Rightarrow\quad T^\perp\subseteq S^\perp. $$ Since $M$ and $N$ are both contained in $M+N$, it folows that $ (M+N)^\perp \subseteq M^\perp$ and $ (M+N)^\perp \subseteq N^\perp$. Hence $ (M+N)^\perp \subseteq M^\perp\cap N^\perp$.