Question on isometries

Dear Professor,

Firstly thanks for taking the time out to reply to our messages, we really appreciate the help and guidance from yourself. I wanted to ask the following short question about the 2201 notes:

On page 56 Thm 5.4.150: after showing that $(i)\Rightarrow (ii)$ and $(ii) \Rightarrow (iii)$, can’t we simply show $(iii) \Rightarrow (i)$ by saying:

$||T^*Tu||=||u||$

and proceeding exactly as the last line of the proof at the bottom?

$\langle (T^*T-I)u, v \rangle=0$
for *all* $v$, and then one sets $v= (T^*T-I)u$. This wouldn’t be admissible if we just had $\langle (T^*T-I)u, u\rangle=0$. Try it out to see clearly for yourself what I mean.