Operator Norm

Let $V$ be a complex vector space equipped with a norm $\lVert\cdot\rVert$, let $T$ be a bounded linear operator on $V$, and let $c$ be a real number.

The number $c$ is an operator norm of $T$ if $c$ is a bound for $T$ and

$$c\le C\qquad\text{for every bound }C\text{ for }T,$$

the order being that of the ordered field of real numbers.