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I am sorry for the trivial question, but I am a little bit confused about this notation in literature. Let $H_1$ and $H_2$ be two Hilbert spaces. I am interested in understanding what means that an operator $A$ is bounded from $H_1$ to $H_2$, i.e. $A\in\mathscr{B}(H_1, H_2)$. It means that taken $u\in H_1$ we have $$\Vert Au\Vert_{H_2}\leq C\Vert A\Vert \Vert u\Vert_{H_1}?$$ Or it means that taken $u\in H_2$ we have $$\Vert Au\Vert_{H_1}\leq C\Vert A\Vert \Vert u\Vert_{H_2}?$$

Thank you in advance!

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    $\begingroup$ $u\in H_1$ and $Au\in H_2,$ so norms there (the former possibility you gave) $\endgroup$ May 25, 2020 at 8:26

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Let $T:H_1 \rightarrow H_2$ be an operator. This operator is bounded if: $$ \| Tv \|_{H_2} \leq M \| v \|_{H_1} $$ for some constant $M >0$. The "best" constant, namely: $$ \| T \| = \sup_{v \in H_1} \frac{\| Tv \|_{H_2}}{\| v \|_{H_1}} $$ is called the norm of $T$.

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