Definition:Norm/Bounded Linear Functional/Inner Product Space/Definition 3

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

Let $\struct {V, \innerprod \cdot \cdot}$ be an inner product space over $\mathbb F$ with $V \ne \set 0$.

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

Let $\norm \cdot$ be the inner product norm for $\struct {V, \innerprod \cdot \cdot}$.

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

As $L$ is bounded, it is assured that $\norm L < \infty$.

Also see

 * Equivalence of Definitions of Norm of Linear Functional

Special cases

 * Definition:Norm on Bounded Linear Transformation, of which this is a special case.