DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 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 space $C_{0} (R^{n})$ of continuous functions vanishing at infinity

Definition

Let $C_{0} (R^{n})$ be the space of continuous functions $f : R^{n} \to R$ such that for every $ε > 0$ there is a compact set $K \subseteq R^{n}$ with

∣ f (x) ∣ < ε (x \notin K) .

Equivalently, every $ε$ -tail of $f$ is eventually small outside a large Euclidean ball; this is the correct $L^{\infty}$ -closure target of $C_{c} (R^{n})$ .

Depends on

Used by

The L^∞-closure of C_c(ℝⁿ) is C₀(ℝⁿ), not all of L^∞(ℝⁿ) Theorem

Dependency tree · two levels

23 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 space C0(Rn)C0​(Rn) of continuous functions vanishing at infinity