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 $3.2$