False statementConstruction: AI-adaptedVerification: AI-adaptedprecheck passaudited 2026-09-01

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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.
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FALSE: $C_{c} (R^{n})$ is dense in $L^{\infty} (R^{n})$

Statement

False claim. $C_{c} (R^{n})$ is dense in $L^{\infty} (R^{n})$ .

Facts & Assumptions

Given: The $L^{\infty}$ -closure theorem for $C_{c} (R^{n})$ .

[L1]

The closure of $C_{c} (R^{n})$ in $L^{\infty}$ is exactly $C_{0} (R^{n})$ (The $L^{\infty}$ -closure of $C_{c} (R^{n})$ is $C_{0} (R^{n})$ , not all of $L^{\infty} (R^{n})$ ).

Refutation

technique · direct

1.1

The constant function $1$ belongs to $L^{\infty} (R^{n})$ but not to [L1] $C_{0} (R^{n})$ , because it does not vanish at infinity.

2.1

By [L1], every $L^{\infty}$ -limit of compactly supported continuous functions [L1, step 1.1] lies in $C_{0} (R^{n})$ . Therefore $1$ cannot lie in the closure of $C_{c} (R^{n})$ , and the claim is false.

L1step 1.1∎

Depends on

The $L^\infty$-closure of $C_c(\mathbb{R}^n)$ is $C_0(\mathbb{R}^n)$, not all of $L^\infty(\mathbb{R}^n)$

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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)

FALSE: Cc(Rn)Cc​(Rn) is dense in L∞(Rn)L∞(Rn)