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Linear Operations in Hilbert Spaces

Hi, I am having difficulty understanding the proof of Theorem 4.3(b) in the book Linear Operations in Hilbert Spaces by Weidmann. Can someone please explain the 3 statements $$ \|Tf\|\geq |\langle Tf,g\rangle | \text{ for all $g$ with } \|g\|=1, $$ $$ \exists g_n:\|g_n\|=1, g_n\rightarrow \|Tf\|^{-1}Tf, $$ and $$ |\langle Tf,g_n \rangle |\rightarrow \|Tf\|. $$

Thanks.

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Statement 1: This is just Cauchy Schwarz

Statement 2: This follows from the facts that $R(T) \subseteq \overline{M}$ and $Tf \in R(T)$ (and thus $||Tf||^{-1}Tf \in R(T)$)

Statement 3: This follows from inner producting both sides of statement 2 with $Tf$.

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  • $\begingroup$ Oh, I should have recognized Cauchy Schwarz... Now everything's clear, thanks. $\endgroup$ – nguyen Jul 3 '17 at 10:12

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