Bound for a Linear Operator and Bounded Linear Operator

Let $V$ be a complex vector space equipped with a norm $\lVert\cdot\rVert$, let $T$ be a linear operator on $V$, and let $C$ be a real number with $0\le C$, the order being that of the ordered field of real numbers.

The number $C$ is a bound for $T$ if

$$\lVert T(u)\rVert\le C\,\lVert u\rVert\qquad\text{for every }u\in V.$$

The operator $T$ is bounded if some real number $C$ with $0\le C$ is a bound for $T$.