T is an operator on a finite dimensional inner product space. How do I show that if T is a projection, then it must be an orthogonal projection?
I know I have to use the fact that $T^*T=TT^*$, and I what I want to do is to show that $||T(x)||<||x||$ for all $x\in V$ with the inner product, but I don't know how to get there.