I just copy it from my notes :

Let $M$ be a closed subspace of Hilbert space $H$ and $P$ be the orthonormal projection from $H$ onto $M$. Let $T : Y \rightarrow Z$ be a bounded linear operator, where $Z$ is a normed space. Show $|| T \circ P || = ||T|| $

First, I'm not sure how it is well - defined since the domain doesn't match. Second, if it is well - defined, how to prove the statement? It is easy to show $$|| T \circ P || \le ||T|| $$ since $$|| T \circ P || \le ||T||||P|| = ||T|| $$ if $|P| \neq0$. How can the reverse direction be shown?

Thank you!


This is not true: if $M=\{0\}$ then $P=0$, $\|P\|=0$, $T\circ P=0$...


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.