Definition:Norm/Bounded Linear Functional/Normed Vector Space/Definition 3

Definition
Let $\mathbb F$ be a subfield of $\C$.

Let $\struct {V, \norm \cdot}$ be a normed vector space over $\mathbb F$ with $V \ne \set 0$.

Let $L : V \to \mathbb F$ be a bounded linear functional.

The norm of $L$ is defined as the supremum:
 * $\norm L = \ds \sup \set {\frac {\size {L v} } {\norm v}: v \in V, v \ne \bszero_V}$