Definition:Norm/Bounded Linear Functional/Definition 3
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Definition
Let $\mathbb F$ be a subfield of $\C$.
Let $\struct {V, \norm {\, \cdot \,} }$ be a normed vector space.
Let $L : V \to \mathbb F$ be a bounded linear functional.
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.
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 $3.3$