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.

Dictionary of the classical $L^{p}$ and $ℓ^{p}$ Banach spaces

Example

The standard Banach-space dictionary is:

$L^{p} (μ)$ is Banach for every $1 \leq p \leq \infty$ ;
$ℓ^{p}$ is the counting-measure instance of $L^{p}$ ;
$c_{00}$ is dense in $ℓ^{p}$ for $1 \leq p < \infty$ and dense in $c_{0}$ for the supremum norm.

Facts & Assumptions

Given: The classical spaces listed above.

[L1]

The classical $L^{p}$ spaces are Banach (The classical $L^{p}$ spaces are Banach spaces).

[L2]

The example of $c_{00}$ records its dense embeddings into $c_{0}$ and $ℓ^{p}$ (The finitely supported sequences form an incomplete normed space with different standard completions).

Verification

technique · direct

1.1

The first two bullets are exactly the content of [L1].

1.2

The third bullet is exactly the content of [L2].

2.1

Combining steps 1.1 and 1.2 gives the stated dictionary.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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

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

Dictionary of the classical LpLp and ℓpℓp Banach spaces