1
$\begingroup$

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 $||T||=\sup\{||Tv||_H: ||v||_H=1\}$.

Thanks for your help.

  • 0
    You are right. Thanks.2012-12-02

1 Answers 1

3

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\| $