Bound for a Linear Operator and Bounded Linear Operator

Let $V$ be a \reftext{def:vector-space-2026a}{complex vector space} equipped with a \reftext{def:complex-normed-space-2026a}{norm} $\lVert\cdot\rVert$, let $T$ be a \reftext{def:linear-operator-2026a}{linear operator} on $V$, and let $C$ be a \reftext{def:real-numbers-c54-2026c}{real number} with $0\le C$, the order being that of the \reftext{def:ordered-field-c54-2026b}{ordered field} of real numbers.

The number $C$ is a \textbf{bound} for $T$ if

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