DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-31

How statement and proof provenance work

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.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Conjugate exponents, including the endpoint conventions

Definition

Numbers $p, q \in [1, \infty]$ are conjugate exponents when

$\frac{1}{p} + \frac{1}{q} = 1,$

with the convention $1 / \infty : = 0$ . Thus the endpoint pairs $(1, \infty)$ and $(\infty, 1)$ are conjugate, and if $1 < p < \infty$ then the conjugate of $p$ is

$q = \frac{p}{p - 1} .$

No other endpoint pair is conjugate: $(1, 1)$ gives $2$ , while $(\infty, \infty)$ gives $0$ .

Used by

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

John K. Hunter, Measure Theory, Definition 7.6 (standard reference, not scraped)
Sheldon Axler, Measure, Integration & Real Analysis, Section 7A (standard reference, not scraped)