ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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The functional $f \mapsto \int_{0}^{1 / 2} f$ on $L^{p} [0, 1]$ has norm $2^{- 1 / q}$

Example

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Let $1 \leq p < \infty$ and let $q$ be conjugate to $p$ . On $[0, 1]$ with Lebesgue measure, define $Λ ([f]) : = \int_{0}^{1 / 2} f (x) d x .$ Then $Λ$ is a bounded linear functional on $L^{p} [0, 1]$ and $∥ Λ ∥ = 2^{- 1 / q},$ with the endpoint convention $2^{- 1 / \infty} = 1$ when $p = 1$ .

Facts & Assumptions

Given: The Axiom of Countable Choice and an exponent $1 \leq p < \infty$ with conjugate exponent $q$ .

[L1]

The interval $[0, 1 / 2]$ has Lebesgue measure $1 / 2$ (A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included).

[L2]

If $g \in L^{q} (μ)$ , then the pairing functional $Λ_{g}$ has norm $∥ g ∥_{q}$ in the ranges treated on the A page (The functional $Λ_{g}$ has norm $∥ g ∥_{q}$ ; for $q = \infty$ assume $μ$ is semifinite).

Verification

technique · Represent the functional by the indicator of $[0,1/2]$, compute that indicator's $L^q$ norm from its measure, and invoke the norm formula for $\Lambda_g$

1.1

Let $g : = 1_{[0, 1 / 2]}$ . [given, construct] Then $Λ ([f]) = \int_{0}^{1} f (x) g (x) d x = Λ_{g} ([f]),$ so $Λ$ is exactly the pairing functional associated to $g$ .

givenconstruct

1.2

By [L1], if $q < \infty$ then $∥ g ∥_{q}^{q} = \int_{0}^{1} ∣ g ∣^{q} d x = \int_{0}^{1 / 2} 1 d x = \frac{1}{2},$ so $∥ g ∥_{q} = 2^{- 1 / q} .$

L1given

1.3

If $q = \infty$ , then $∣ g ∣ \leq 1$ everywhere and $g = 1$ on a set of positive measure, so $∥ g ∥_{\infty} = 1 = 2^{- 1 / \infty}$ .

L1given

2.1

Applying [L2] to step 1.1 and steps 1.2-1.3 gives $∥ Λ ∥ = ∥ Λ_{g} ∥ = ∥ g ∥_{q} = 2^{- 1 / q} .$

L2step 1.1step 1.2step 1.3∎

Depends on

Used by

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Dependency tree · two levels

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Sources

Gerald B. Folland, Real Analysis, 2nd ed., Section 6.2 (standard reference, not scraped)
John K. Hunter, Measure Theory, Proposition 7.13 (standard reference, not scraped)

The functional f↦∫01/2ff↦∫01/2​f on Lp[0,1]Lp[0,1] has norm 2−1/q2−1/q