# Projection

• October 18th 2009, 03:46 AM
problem
Projection
Let $H$ be a Hilbert Space and $P$ and $Q$ be the projections on closed linear subspaces $M$ and $N$ of $H$.
Prove that if $Q-P$ is a projection,then its range is $N\cap M^\perp$.

My attempt:
Let $x\in$ range of $Q-P.$
Since $Q-P$ is a projection,
$\Rightarrow(Q-P)x=x$
$\Rightarrow Qx-Px=x$
I am stuck here as i don't know how to proceed to show that $x\in N$ and $x\in M^\perp$

Can anyone give me some hints to proceed?
If I am in the wrong way to prove the question,please tell me.(Happy)