Hmm. Interesting problem. The trick is that applying the projection operator twice should be the same as applying it just once . Couldn't you multiply each by the scalar
So,
Let and for :
.
For any , find its projection on .
I have no idea where to start. I know that the projection map has certain properties (linear, self-adjoint, operator norm of 1, whatever), but is there any specific formula to find it?