TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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.

Compactly supported distributions are tempered

Statement

Every compactly supported distribution $v \in D^{'} (R^{n})$ has a unique extension $\tilde{v} \in S^{'} (R^{n})$ . Restricting $\tilde{v}$ to $D$ recovers $v$ . No choice axiom is required.

Facts & Assumptions

Given: A distribution $v$ with compact support in $R^{n}$ .

[F1]

Such a distribution extends uniquely to a continuous linear functional on $C^{\infty}$ , with $\tilde{v} (f) = v (χ f)$ for any fixed cutoff equal to one near the support (Compactly supported distributions extend to smooth functions).

[F2]

A finite Schwartz-seminorm estimate proves temperateness (Finite seminorm bound characterizes tempered distributions).

[F3]

The inclusion $D \to S$ is continuous with dense image (Test function inclusion in schwartz space is continuous).

Proof

technique · cutoff extension and density

1.1

Restrict the $C^{\infty}$ extension from [F1] to $S$ . Its continuity gives a compact $K$ , an integer $m$ , and $C \geq 0$ satisfying the following estimate.

$∣ \tilde{v} (φ) ∣ \leq C \max_{∣ β ∣ \leq m} \sup_{x \in K} ∣ \partial^{β} φ (x) ∣ \leq C \max_{∣ β ∣ \leq m} p_{0, β} (φ) .$

Thus the restriction is tempered by [F2]. [F1, F2]

1.2

For $ψ \in D$ , the extension property in [F1] gives $\tilde{v} (ψ) = v (ψ)$ ; this includes the zero distribution and empty support. Hence $\tilde{v}$ really extends $v$ .

2.1

If $w \in S^{'}$ is another extension, then $w - \tilde{v}$ vanishes on $D$ . This difference is continuous on $S$ , and $D$ is dense there, so it vanishes on all of $S$ . Therefore the extension is unique.

F3step 1.2∎

Depends on

Used by

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

Semyon Dyatlov, Lecture notes for 18.155 (2022) (standard reference, not scraped)
Radu Gelca, Functional Analysis (standard reference, not scraped)