# Definition:Norm/Bounded Linear Functional

*This page is about the norm on a bounded linear functional. For other uses, see Norm.*

## Definition

Let $\GF$ be a subfield of $\C$.

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

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

### Definition 1

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

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

### Definition 2

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

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

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.

### Definition 3

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.

### Definition 4

The **norm** of $L$ is the infimum:

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

## Also known as

Since this is the norm associated with the normed dual space, it is often known as the **dual norm**.

## Also see

- Norm on Bounded Linear Functional is Finite
- Corollary of Equivalence of Definitions of Norm of Linear Functional where it is shown, for all $v \in V$, that $\size {L v} \le \norm L \norm v$.

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

## 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 $3.2$