Definition:Norm/Bounded Linear Functional/Definition 3

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

Let $\struct {V, \norm {\, \cdot \,} }$ be a normed vector space.

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

The norm of $L$ is defined as the supremum:


 * $\norm L = \sup \set {\dfrac {\size {L v} } {\norm v}: v \in V, v \ne \bszero_V}$

where the supremum is taken in $\struct {\closedint 0 \infty, \le}$ where $\le$ is the restriction of the standard ordering of the extended real numbers.