False statementConstruction: Literature-sourcedVerification: AI-adaptedPipeline-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.

Every positive linear functional on C_c(X) is uniformly sup-norm bounded

Statement

The assertion that every positive linear functional on $C_{c} (X)$ is bounded for the global supremum norm is false.

Facts & Assumptions

Given: $X = R$ and $Λ (f) = \int_{R} f d λ$ .

Refutation

technique · counterexample

1.1

The functional is linear and positive on $C_{c} (R)$ . For each $n$ , choose a cutoff $f_{n} \in C_{c} (R)$ with $0 \leq f_{n} \leq 1$ and $f_{n} = 1$ on $[- n, n]$ .

given

2.1

Then $∥ f_{n} ∥_{\infty} = 1$ but $Λ (f_{n}) \geq 2 n$ . No constant $C$ can satisfy $∣ Λ (f) ∣ \leq C ∥ f ∥_{\infty}$ for every $f \in C_{c} (R)$ .

step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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

Donald L. Cohn, Measure Theory, 2nd ed., Chapter 7 (standard reference, not scraped)