DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-01

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A bounded linear functional on $L^{p} (μ)$ and its operator norm

Definition

Let $(X, A, μ)$ be a measure space and let $1 \leq p < \infty$ . A map $Λ : L^{p} (μ) \to R,$ is a bounded linear functional when it is linear and there exists $C \in [0, \infty)$ such that $∣ Λ ([f]) ∣ \leq C ∥ [f] ∥_{p} ([f] \in L^{p} (μ)) .$

Its operator norm is $∥ Λ ∥ : = \sup {∣ Λ ([f]) ∣ : ∥ [f] ∥_{p} \leq 1} .$

Because $L^{p} (μ)$ is a normed space by The $L^{p}$ norm descends to the quotient and makes $L^{p}$ a normed space for $1 \leq p \leq \infty$ , the unit ball is nonempty and the displayed supremum is over a well-defined subset of $[0, \infty]$ . When $Λ$ is bounded, the defining inequality shows $∥ Λ ∥ \leq C$ .

Depends on

Used by

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Gerald B. Folland, Real Analysis, 2nd ed., Section 6.2 (standard reference, not scraped)
Richard F. Bass, Real Analysis for Graduate Students, Section 15.4 (standard reference, not scraped)
John K. Hunter, Measure Theory, Section 7.5 (standard reference, not scraped)

A bounded linear functional on Lp(μ)Lp(μ) and its operator norm