My question is:
Let $H$ be a Hilbert space and $T \in B(H)$. Prove that $T$ is a projection if and only if $T$ is the identity on the orthogonal complement of its kernel.
Thanks
My question is:
Let $H$ be a Hilbert space and $T \in B(H)$. Prove that $T$ is a projection if and only if $T$ is the identity on the orthogonal complement of its kernel.
Thanks
If "projection" is changed to "orthogonal projection", then here are some hints.
Hint 1: $\ker(T)^\perp=\{u\in H:Tx=0\Rightarrow \langle u,x\rangle=0\}$
Hint 2: Any $v\in H$ can be written as $v=u+k$ where $u\in\ker(T)^\perp$ and $k\in\ker(T)$.