Category:Definitions/Norm on Bounded Linear Functional
This category contains definitions related to Norm on Bounded Linear Functional.
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:
- $\norm L = \inf \set {c > 0: \forall v \in V : \size {L v} \le c \norm v}$
Pages in category "Definitions/Norm on Bounded Linear Functional"
The following 5 pages are in this category, out of 5 total.