I'm working on a different problem, but I encountered this question, which would basically solve my problem for me.
The problem I'm working on requires me to show that $\langle P_Mx,y\rangle=\langle P_M x,P_M y\rangle=\langle x,P_My\rangle,$ where $P$ is the projection operator onto a closed subspace, $M$, of an inner product space. If projections do in fact preserve norms, then I can just use the polarization identity along with the fact that projections are idempotent.
But I feel like I have to be misunderstanding something. The claim that $T$ is orthogonal iff $||Tx||^2=||x||^2$ seems patently false. For example, if $x=(1,1)\in\mathbb{R}^2$, and we let $T$ be the projection onto the $x$ axis, then $Tx = (1,0)$. So $||Tx||^2=1$, but $||x||^2 = 2$.
Any clarification on what I'm missing would be helpful.
Thanks in advance
Unless you are projecting onto the whole space they do not preserve the norm. What you need is decompose $y$ as $$ Y=P_M y+ P_{M^\perp }y $$ In your first expression. When you distribute the product the second piece vanishes because it is an inner product of vectors in orthogonal subspaces so you get your second term etc.