LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02

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 uniform closure of a unital real function algebra is closed under absolute value, maximum, and minimum

Statement

If $A\subseteq C(K,\mathbb R)$ is a unital real function algebra and $\overline A$ is its uniform closure, then $u\in\overline A$ implies $|u|\in\overline A$ ; consequently $u\vee v$ and $u\wedge v$ lie in $\overline A$ whenever $u,v$ do.

Facts & Assumptions

Given: A unital real function algebra $A$ and $u,v\in\overline A$ .

[L1]

The algebra operations and constants are those in A unital point-separating real subalgebra of $C(K,\mathbb R)$ .

[L2]

Polynomials uniformly approximate $|t|$ on every bounded closed interval (Polynomials are uniformly dense in $C([a,b],\mathbb R)$ for every closed interval).

Proof

technique · direct

1.1

Choose $a_n\in A$ converging uniformly to $u$ . Their ranges, together with that of $u$ , lie in one bounded interval.

givenchoose

1.2

By [L2], choose polynomials $p_j$ with $p_j(0)=0$ converging uniformly to $|t|$ on that interval. Then $p_j(a_n)\in A$ by [L1].

L1L2choose

2.1

A diagonal choice of $j,n$ makes $p_j(a_n)$ uniformly converge to $|u|$ , so $|u|\in\overline A$ .

step 1.1step 1.2algebra

3.1

The identities $u\vee v=(u+v+|u-v|)/2$ and $u\wedge v=(u+v-|u-v|)/2$ and [L1] give the remaining closure.

step 2.1L1algebra∎

Depends on

Used by

Real Stone--Weierstrass theorem for compact metric spaces Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 53 results over 14 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

The Stone--Weierstrass Theorem and its Applications (standard reference, not scraped)