Definition:Norm/Bounded Linear Functional/Definition 1

Definition

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

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

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

$\norm L = \sup \set {\size {L v}: \norm v \le 1}$

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $\text{I}$ Hilbert Spaces: $\S 3.$ The Riesz Representation Theorem: Definition $3.2$