TheoremStatement: 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.

Real Stone--Weierstrass theorem for compact metric spaces

Statement

Let $K$ be a nonempty compact metric space and let $A\subseteq C(K,\mathbb R)$ be a unital subalgebra which separates points. Then $A$ is dense in $C(K,\mathbb R)$ for the supremum metric.

Facts & Assumptions

Given: $f\in C(K,\mathbb R)$ and $\varepsilon>0$ .

[L1]

The uniform closure $\overline A$ is closed under pointwise maximum and minimum (The uniform closure of a unital real function algebra is closed under absolute value, maximum, and minimum).

[L2]

For distinct $x,y\in K$ and prescribed real values at $x,y$ , $A$ contains a function taking those two values (A unital separating real function algebra interpolates arbitrary values at two distinct points).

[L3]

Every open cover of the compact metric space $K$ has a finite subcover (Open cover, subcover, compact metric space, and compact subset of a metric space).

Proof

technique · constructive

1.1

For fixed $x,y\in K$ , choose $u_{x,y}\in A$ with $u_{x,y}(x)=f(x)$ and $u_{x,y}(y)=f(y)$ , using $f(x)\mathbf1$ when $x=y$ and [L2] otherwise.

L2algebra

2.1

For fixed $x$ , the open sets $U_{x,y}=\{z:u_{x,y}(z)>f(z)-\varepsilon\}$ cover $K$ , since $y\in U_{x,y}$ . Select $y_1,\ldots,y_r$ whose sets cover $K$ by [L3].

step 1.1L3construct

3.1

Put $g_x=\max_j u_{x,y_j}\in\overline A$ . Then $g_x(x)=f(x)$ and $g_x>f-\varepsilon$ on $K$ .

L1step 2.1algebra

4.1

The open sets $V_x=\{z:g_x(z)<f(z)+\varepsilon\}$ cover $K$ . By [L3], choose $x_1,\ldots,x_s$ whose $V_{x_i}$ cover $K$ .

step 3.1L3construct

5.1

The function $h=\min_i g_{x_i}$ belongs to $\overline A$ by [L1], and $f-\varepsilon<h<f+\varepsilon$ pointwise. Thus $\lVert h-f\rVert_\infty<\varepsilon$ .

L1step 3.1step 4.1algebra

6.1

Since $\varepsilon$ was arbitrary and every $h$ of step 5.1 lies in the closed set $\overline A$ , the function $f$ lies in $\overline A$ . Thus $\overline A=C(K,\mathbb R)$ and $A$ is dense.

step 5.1discharge-construct∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 43 results over 13 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

Stone-Weierstrass Theorem (University of Chicago) (standard reference, not scraped)