One paragraph in my text is to prove that $\|T\|=\sup\{|\langle Tf, g\rangle|:\|f\|<1, \|g\|<1\}$, where we have a bounded linear operator between two Hilbert spaces $T:\mathcal H_1\rightarrow \mathcal H_2$ and $f\in\mathcal H_1, f\in\mathcal H_2$. FYI, the norm of $T$ is defined as the minimum constant such that $\|Tf\|_{\mathcal H_2}\leq M\|f\|_{\mathcal H_1}$. I can following the details in my book. However, I do not understand the rationale behind the proof. Any clarification will be appreciated.
Normally, if we want to show $a=b$, we can proceed to show $a\geq b$ and $a\leq b$. However, the proof in my book proceeded as follows.
- Assume $\|T\|\leq M$, then show $|\langle Tf, g\rangle|\leq M$. Update: This makes sense if we rephrase the proof as follows. If $|\langle Tf, g\rangle|\leq M$ for all $M$, then it is clear that $\sup|\langle Tf, g\rangle|\leq \inf M=\|T\|.$
- Assume $\sup\{|\langle Tf, g\rangle|:\|f\|<1, \|g\|<1\}\leq M$, then show $\|Tf\|_{\mathcal H_2}\leq M\|f\|_{\mathcal H_1}$.
My question is why proving 1 and 2 above is sufficient to prove the opening lemma.
(1) proves that $|\langle Tf,g \rangle|\leq ||T||$, so $\sup|\langle Tf, g\rangle|\leq ||T||$.
(2) similarly proves that $||T||\leq \sup|\langle Tf, g\rangle|$, by taking $M=\sup|\langle Tf, g\rangle|\leq ||T||$.
What's going on here is you're trying to prove a sup of one set of numbers is the same as the sup of another set of numbers. If you think of it in terms of least upper bounds you would arrive at trying to prove the statements with "$\leq M$".