ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-22

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.

Character space of C(K)

Example

Assume the Axiom of Dependent Choice (The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain). Let $K$ be a nonempty compact Hausdorff space. Then the evaluation map

$e : K ⟶ Δ (C (K)), e (x) : = {e v}_{x}, {e v}_{x} (f) = f (x),$

is a homeomorphism onto the character space of the complex Banach algebra $C (K) = C (K, C)$ with the supremum norm. In particular, for $K = [0, 1]$ the characters of $C ([0, 1])$ are exactly the evaluations $f \mapsto f (t)$ at points $t \in [0, 1]$ , and each occurs exactly once.

Facts & Assumptions

Given: Dependent Choice, a nonempty compact Hausdorff space $K$ , and the algebra $C (K)$ with the supremum norm and pointwise operations.

[L1]

For a nonempty compact Hausdorff space, every character of $C (K)$ is an evaluation at a unique point and the evaluation map is a homeomorphism onto $Δ (C (K))$ (Characters of continuous functions are evaluations, The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain).

[L2]

$[0, 1]$ is a nonempty compact Hausdorff space with its subspace topology from $R$ . [algebra]

Verification

technique · direct

1.1

By [L1] the map $e$ is a homeomorphism for every nonempty compact Hausdorff $K$ , so in particular every character of $C (K)$ is ${e v}_{x}$ for a unique $x$ , and the topology on $Δ (C (K))$ is the transported topology of $K$ ; no computation beyond [L1] is needed.

2.1

For $K = [0, 1]$ the hypotheses of [L1] hold by [L2], so the description of the characters of $C ([0, 1])$ follows; explicitly, ${e v}_{s} = {e v}_{t}$ would give $f (s) = f (t)$ for all continuous $f$ , and the coordinate function $f (u) = u$ then forces $s = t$ .

1.1L2algebra∎

Remarks

Only the character space is asserted here. Under Dependent Choice the supplied evaluation lemma identifies $Δ (C (K))$ with $K$ ; this example does not claim the stronger, AC-dependent correspondence with all maximal ideals.
Dependent Choice is inherited from the Urysohn input of [L1] and is used only there.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

15 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

Vahid Shirbisheh, Lectures on C-star Algebras, v2 — Remark 3.1.36 and §3.1, printed pp. 54–67 (standard reference, not scraped)
Theo Bühler and Dietmar A. Salamon, Functional Analysis — §5.5.1, printed pp. 258–267 (standard reference, not scraped)