RemarkRemark: Literature-sourcedProof: Not supplied‡ not proved here

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.

‡ Recorded, not proved here. This statement is included so the library can refer to it honestly, with a citation to the literature. It is not proved anywhere in this library: the track that would prove it has not been developed here yet.

The $p = \infty$ case is recorded but not proved here

Remark

For $p = \infty$ , the concrete map $g ⟼ Λ_{g}, Λ_{g} (f) : = \int f g d μ,$ from $L^{1} (μ)$ into the bounded linear functionals on $L^{\infty} (μ)$ is not surjective in general. This page therefore stops at $1 \leq p < \infty$ and records the failure of the $p = \infty$ analogue without proving it here.

The boundary witness belongs to the library's recorded-not-proved functional analysis seam The dual of $ℓ^{\infty}$ is $b a (2^{N})$ , the finitely additive charges ‡. On $[0, 1]$ the standard route extends point evaluation from a smaller subspace by Hahn-Banach, which this page does not use.

Used by

Orientation only: the dual-space phrasing of the concrete Lᵖ theorem Remark

Dependency tree · 0 levels

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

Sources

Gerald B. Folland, Real Analysis, 2nd ed., paragraph after Theorem 6.15 (standard reference, not scraped)
John K. Hunter, Measure Theory, paragraph after Theorem 7.14 (standard reference, not scraped)

The p=∞p=∞ case is recorded but not proved here

Remark

Used by

Dependency tree · 0 levels

Sources

The $p = \infty$ case is recorded but not proved here