Let P be a projector.

Show that if $\displaystyle \left\|P \right\|_2 = 1, $ then P is an orthogonal projection.

I know that for any projection P, $\displaystyle \left\|P \right\|_2 \geq 1, $ , and that if P is an orthogonal projection, then $\displaystyle \left\|P \right\|_2 = 1, $, but I am not sure how to prove the other way around.