I'm a little surprised that I'm stuck on this point, but I have my bad days. I am trying to understand a statement made in the proof of Theorem VI.1 in p.184 of Reed and Simon Volume I Functional Analysis. Let $L(H,\mathbb{C})$ denote the space of linear functionals (operators) from the Hilbert space $H$ to another Hilbert space of complex numbers $\mathbb{C}$. Now, Let $T$ be an operator in $L(H,H)$, the space of operators from $H$ into $H$. Then consider $Tx$ in $L(H,\mathbb{C})$.
It says the operator norm of $Tx$ in $L(H,\mathbb{C})$ is the same as the norm of $Tx$ in $H$.
I know what an operator norm is. To define the operator norm of $Tx$ in $L(H,\mathbb{C})$:
$\sup_{y\neq 0} \frac{|\langle Tx,y \rangle|}{\langle y,y \rangle^{1/2}}$
and the norm of $Tx$ in H is just $\langle Tx,Tx \rangle^{1/2}$.
The conclusion of the two being equal tells me that assuming $\langle y,y \rangle=1$, we have:
$|\langle Tx,y \rangle|= \langle Tx,Tx \rangle^{1/2}$
Is that even true? Where did I go wrong?!