RemarkRemark: AI-adaptedProof: 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.

$L^{p}$ completeness and the Banach-property wording

Riesz-Fischer completeness of $L^{p}$ for $1 \leq p \leq \infty$ proves the mathematics this page needs: with the descended norm, $L^{p} (μ)$ is complete for $1 \leq p \leq \infty$ . Later functional-analysis pages supply the vocabulary seam by naming a complete normed space a Banach space. The terminology is deferred; the completeness proof is not.

Depends on

Riesz-Fischer completeness of $L^p$ for $1 \le p \le \infty$

Used by

Nothing in the library uses this result yet.

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

John K. Hunter, Measure Theory, Theorem 7.10 (standard reference, not scraped)
Sheldon Axler, Measure, Integration & Real Analysis, Theorem 7.24 (standard reference, not scraped)

LpLp completeness and the Banach-property wording

Depends on

Used by

Dependency tree · two levels

Sources

$L^{p}$ completeness and the Banach-property wording