ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 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.

$C (K)$ is Banach when $K$ is compact metric

Example

If $K$ is a nonempty compact metric space, then the space $C (K)$ of continuous scalar functions on $K$ , with the supremum norm, is a Banach space.

Facts & Assumptions

Given: A nonempty compact metric space $K$ .

[L1]

Every bounded continuous function space $C_{b} (X)$ is Banach for the supremum norm ( $C_{b} (X)$ is Banach for the supremum norm).

[L2]

A continuous real-valued function on a nonempty compact metric space is bounded and attains its extrema (A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value).

Verification

technique · direct

1.1

If $f \in C (K)$ , [L2] makes $f$ bounded. Hence $C (K) = C_{b} (K)$ as sets and the supremum norms agree.

2.1

Since $K$ is nonempty and compact, [L1] makes $C_{b} (K)$ Banach for the supremum norm. By step 1.1, this is exactly $C (K)$ with the same norm.

step 1.1L1∎

Depends on

Used by

Polynomials are not complete in the supremum norm on a compact interval Example

Dependency tree · two levels

26 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

Gerald Teschl, Topics in Real and Functional Analysis (standard reference, not scraped)
Theo Buhler and Dietmar A. Salamon, Functional Analysis (standard reference, not scraped)

C(K)C(K) is Banach when KK is compact metric