If P is a projector that is neither 0 or the identity, we prove that \(\displaystyle \left\|P \right\| = \left\| I-P \right\|\)

in any norm induced by a vector norm generated by some inner product.

Do we have that Range(P) = orthogonal complement of [Null(P*)]= orthogonal complement of [Range(I-P*)] ? (where P* is the transpose conjugate of P)

and if yes, can we use this fact to prove the problem?

I have a thought but I am not sure if it will help or go anywhere.

\(\displaystyle <I-P, I-P>=<I,I>-2<I,P>+<P,P>\)

\(\displaystyle -1\leq \frac{<I,P>}{\left\|P \right\|\left\|I \right\|}\leq 1\)

\(\displaystyle <I,I>-2<I,P>+<P,P>=<I,I>+<P,P>\)

I don't know if any of this rambling helps but check out.