ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 2026-09-05

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The abstract norming-functional theorem agrees with the concrete L^p-L^q formula

Example

Let $(X, A, μ)$ be a measure space, let $1 \leq p < \infty$ , and let $q$ be conjugate to $p$ . Assume either $1 < p < \infty$ , or $p = 1$ and $μ$ is sigma-finite. For every nonzero $f \in L^{p} (μ)$ , the abstract Hahn-Banach theorem produces a unit-norm functional $Φ \in (L^{p} (μ))^{*}$ with $Φ (f) = ∥ f ∥_{p}$ , and the earlier $L^{p} - L^{q}$ duality page realizes the same value concretely by pairing against the usual $L^{q}$ extremizer.

Facts & Assumptions

Given: A measure space $(X, A, μ)$ , an exponent $1 \leq p < \infty$ with conjugate exponent $q$ , and a nonzero element $f \in L^{p} (μ)$ in one of the ranges covered by the $L^{p} - L^{q}$ duality page.

[L1]

Every nonzero vector has a norming functional (Every nonzero vector has a norming functional).

[L2]

In the same $L^{p} - L^{q}$ ranges, the $L^{p}$ norm is the supremum of pairings against unit $L^{q}$ functions (The $L^{p}$ norm is the supremum of pairings against unit $L^{q}$ functions).

Verification

technique · direct

1.1

Apply [L1] to the nonzero vector $f \in L^{p} (μ)$ . This gives a functional $Φ \in (L^{p} (μ))^{*}$ with $∥ Φ ∥ = 1$ and $Φ (f) = ∥ f ∥_{p}$ .

L1given

1.2

By [L2], the same number $∥ f ∥_{p}$ is obtained as $\sup {∣ \int f g d μ ∣ : ∥ g ∥_{q} \leq 1} .$ In the standard explicit realization, one takes $g = ∣ f ∣^{p - 1} sgn f / ∥ f ∥_{p}^{p - 1}$ when $1 < p < \infty$ , and $g = sgn f$ when $p = 1$ .

L2givenalgebra

2.1

Thus the abstract existence statement from Hahn-Banach and the concrete $L^{p} - L^{q}$ formula from the earlier page identify the same norming phenomenon: one proves that some unit functional attains $∥ f ∥_{p}$ , and the other writes an attaining functional down explicitly.

step 1.1step 1.2∎

Depends on

Used by

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Dependency tree · two levels

8 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, Section 6.2 (standard reference, not scraped)
Richard F. Bass, Real Analysis for Graduate Students, Corollary 15.9 (standard reference, not scraped)