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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!

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

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