I'm studying Reed and Simon's "Methods of Modern Mathematical Physics" Vol. 1 (http://www.math.bme.hu/~balint/oktatas/fun/notes/Reed_Simon_Vol1.pdf). In the proof of the square root lemma (p.196) they use the equation
$\|I-A\|=\sup\limits_{|\varphi|=1}|((I-A)\varphi,\varphi)|,$
where $A$ is a bounded positive operator on a Hilbert space $\mathcal{H}$ and $I$ is the identity operator in $\mathcal{H}$. I understand that for any bounded operator $T$, the Cauchy-Schwarz imequality implies that
$\|T\|\geq\sup\limits_{|\varphi|=1}|(T\varphi,\varphi)|$.
But I am not able to prove the other inequality. Under what hypothesis is it true?