# Inequality of bounded linear operators on Hilbert space

Let $T$ and $S$ be bounded linear operators on a Hilbert space $H$. Verify that: $||TS||\leq ||T||\cdot ||S||$.

The definition of the operator's norm is $||T||=\sup\{||Tv||_H: ||v||_H=1\}$.

Note that for any $x\in H$, $\|Sx\|\leq\|S\|\|x\|$ and similarly $\|Ty\|\leq\|T\|\|y\|$ for any $y\in H$. Thus we have, for any $v$ with $\|v\|=1$, $$\|TSv\|\leq\|T\|\|Sv\|\leq\|T\|\|S\|\|v\|=\|T\|\|S\|$$