DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-01

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 spaces $C_{c} (R^{n})$ and $C_{c}^{\infty} (R^{n})$

Definition

For $n \geq 1$ , let $C_{c} (R^{n})$ be the vector space of continuous functions $f : R^{n} \to R$ with compact support (The support of a function on $R^{n}$ and its compactly supported Riemann integral, Vector-valued functions $f : A \to R^{m}$ , their limits and continuity, with the dictionary to the metric notions).

Let $C_{c}^{\infty} (R^{n})$ be the subspace of those $f \in C_{c} (R^{n})$ that are smooth in the Euclidean multi-index sense ( $C^{k}$ maps and multi-index derivative notation in Euclidean space).

Depends on

Used by

Dependency tree · two levels

31 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

Walter Rudin, Real and Complex Analysis, 3rd ed. (standard reference, not scraped)

The spaces Cc(Rn)Cc​(Rn) and Cc∞(Rn)Cc∞​(Rn)