Projector $P\neq0$ is orthogonal projection if and only if satisfied some of the following mutually equivalent conditions:
a) $P$ is self-adjoint, $P=P^*$
b) $P$ is normal, i.e. $P^*P=PP^*$
c) $P$ is positive
d) $\|P\|=1$
By definition, if $P\neq 0$ is orthogonal projector that is equivalent with a).
$a)\implies b):$ $P^*P=P^2=PP^*$
$a)\iff c):$ $P$ is positive if it's square form is nonnegative. That is equivalent with $\langle Pf,f\rangle$ is real. Operator is self adjoint iff it's square form is real.
Can anyone give me a hint how to prove rest of implications.