RemarkRemark: AI-adaptedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-01‡ rests on unproved material (inherited)

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.

Orientation only: the dual-space phrasing of the concrete $L^{p}$ theorem

Remark

The two concrete representation theorems on this page are exactly what the later functional-analysis dual-space page will summarize as an isometric identification of the real continuous dual of $L^{p} (μ)$ with $L^{q} (μ)$ , under the same hypotheses and with the same endpoint warnings. This page stays with the concrete integral representation until that later track owns the abstract language.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 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

Richard F. Bass, Real Analysis for Graduate Students, sentence after Theorem 15.11 (standard reference, not scraped)
Gerald B. Folland, Real Analysis, 2nd ed., Section 6.2 (standard reference, not scraped)

Orientation only: the dual-space phrasing of the concrete LpLp theorem