Inner Product of Bounded Self Adjoint Linear Operator

Question

$T:V \rightarrow V$ is a bounded self adjoint linear operator, where $V$ is an inner product space.

Why is:

$|<Tv,v>| \leq M \cdot ||v||^2 $

for all $v \in V$, where $M=sup_{||x||=1} \{\mid <Tx,x>\mid \}$

Answer 1

Note that for $v \neq 0$, we have $$ |\langle Tv,v \rangle| = \|v\|^2 \left \langle T\frac{v}{\|v\|},\frac{v}{\|v\|}\right \rangle \leq \|v\|^2 \cdot M $$