Definition:Norm/Bounded Linear Functional/Definition 4

Definition

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

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

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

The norm of $L$ is the infimum:

$\norm L = \inf \set {c > 0: \forall v \in V : \size {L v} \le c \norm v}$

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $\text{I}$ Hilbert Spaces: $\S 3.$ The Riesz Representation Theorem: Proposition $3.3$