inequality regarding norm of linear operator Let $T$ be a linear operator on a vector space $X$. For $x\in X$, I know there's the inequality that $$||Tx||<||T||||x||$$
Yet I'm wondering what are those norms. Are they arbitrary? especially on the right hand side there's operator norm and vector norm, how do we cooperate that?
Thank you!

There are a few equivalent definitions of the operator norm. Perhaps the most relevant for this particular question is the following:
Let $T:V \to W$ be a linear map between two normed vector spaces $V$ and $W$. The operator norm $\|T\|_{op}$ of $T$ is defined to be
$$\|T\|_{op} := \sup \left\{\frac{\|Tx\|}{\|x\|} : x \in V \text{ and } x \not = 0 \right\}$$
It then follows immediately that $\|Tx\| \leq \|T\|_{op} \|x\|$ for any $x \in V$.
Check out Wikipedia for more info.
