RemarkRemark: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

The published Lp duality theorem in abstract notation

Remark

For a sigma-finite measure space and real $L^{p}$ , with $1 \leq p < \infty$ and $1 / p + 1 / q = 1$ (so $q = \infty$ for $p = 1$ ), On a sigma-finite measure space, every bounded linear functional on $L^{p}$ is integration against a unique $L^{q}$ function identifies every bounded functional uniquely as $[f] \mapsto \int f g d μ$ , with norm $∥ g ∥_{q}$ . In the language of The dual space X^* of a normed space and its dual norm, this is a linear isometric identification $(L^{p})^{*} ≅ L^{q}$ . In particular Counting measure specializes the representation theorem to $ℓ^{p}$ and $ℓ^{q}$ at $p = 1$ gives $(ℓ^{1} (R))^{*} ≅ ℓ^{\infty} (R)$ . This remark asserts neither a $p = \infty$ representation nor a complex or arbitrary-measure extension of the cited theorem.

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Sources

Bühler–Salamon, Functional Analysis, Example 1.33, p.33 and Example 1.35, p.36 (standard reference, not scraped)