Calculation of operator norm

Question

$H$ is a Hilbert space, $T: H \to H$ linear bounded operator, $||T||$ is the norm of $T$ given by $$||T||=\sup\{||T(x)||;||x||\le 1 \}. $$

Is it true that $$||T||=\sup\{|\langle Tx,y\rangle|;||x||\le 1 , ||y||\le 1\} ? $$

Answer 1

Have you tried using Cauchy-Schwarz?

$$|\langle Tx, y \rangle | \ \leq \ ||Tx|| \, || y || \ \leq \ ||Tx||$$

Is there a $y$ for which this inequality is an equality?