# (clearification of proof) definition of operator norm

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.

## 1 Answer

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 $Tf$.

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