
Projection
Let $\displaystyle H$ be a Hilbert Space and $\displaystyle P$ and $\displaystyle Q$ be the projections on closed linear subspaces $\displaystyle M$ and $\displaystyle N$ of $\displaystyle H$.
Prove that if $\displaystyle QP$ is a projection,then its range is $\displaystyle N\cap M^\perp$.
My attempt:
Let $\displaystyle x\in$ range of $\displaystyle QP.$
Since $\displaystyle QP$ is a projection,
$\displaystyle \Rightarrow(QP)x=x$
$\displaystyle \Rightarrow QxPx=x$
I am stuck here as i don't know how to proceed to show that $\displaystyle x\in N$ and $\displaystyle x\in M^\perp$
Can anyone give me some hints to proceed?
If I am in the wrong way to prove the question,please tell me.(Happy)