Confusion about definition of orthogonal projection

Question

My professor gave the definition of orthogonal projection as follows : Let H be inner product space. If P $\in \mathbb B(H)$(continuous operator) such that $P^{2}=P$ and ker(P)= (range$P)^\perp$, then it is orthogonal projection. My question is does it is necessary that to include condition P $\in \mathbb B(H)$ as every projection may not be continuous.

Answer 1

The condition $P\in\Bbb B(H)$ is indeed redundant. But not because every projection need to be continuous; it's redundant because any $P$ satisfying the rest of the definition must be continuous.

Indeed, suppose $P:H\to H$ is linear, $P^{2}=P$ and $ker(P)= (range P)^\perp$. For every $x\in H$ we have $$P(x-Px)=0,$$so $$x=y+Px,\quad Py=0.$$Since the range and kernel of $P$ are othogonal this shows that $$||x||^2=||y||^2+||Px||^2\ge||Px||^2,$$so $||P||\le1$.