I am trying to prove $\|T\| \leq \|T\|_{HS}$ I understand everything up until the following two lines, could somebody please explain why
$\| Tx \| \leq \|T \|_{HS}$ $\|x\|$
implies that
$\| T \| \leq \| T \|_{HS}$
It seems to be so trivial, but I lack the insight.