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.

The finitely supported sequences form an incomplete normed space with different standard completions

Example

Let $c_{00} : = {x = (x_{n})_{n \geq 0} : x_{n} = 0 for all but finitely many n} .$ Then $c_{00}$ is an incomplete normed space for the supremum norm, its completion for that norm is $c_{0}$ , and for each $1 \leq p < \infty$ its completion for the $ℓ^{p}$ norm is $ℓ^{p}$ .

Facts & Assumptions

Given: The finitely supported sequence space $c_{00}$ and, for a sequence $x = (x_{n})$ , its truncations $x^{(N)} : = (x_{0}, \dots, x_{N}, 0, 0, \dots)$ .

[L1]

The space $c_{0}$ is Banach for the supremum norm ( $c_{0}$ is Banach for the supremum norm).

[L2]

Any two completions of a normed space are uniquely linearly isometric (Any two completions of a normed space are uniquely linearly isometric).

[L3]

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

Verification

technique · direct

1.1

If $x \in c_{0}$ , then $x^{(N)} \in c_{00}$ and $∥ x - x^{(N)} ∥_{\infty} = \sup_{n > N} ∣ x_{n} ∣ \to 0$ . So $c_{00}$ is dense in $c_{0}$ for the supremum norm.

1.2

For $1 \leq p < \infty$ and $x \in ℓ^{p}$ , the truncations satisfy $∥ x - x^{(N)} ∥_{p}^{p} = \sum_{n > N} ∣ x_{n} ∣^{p} \to 0$ , so $c_{00}$ is dense in $ℓ^{p}$ for the $ℓ^{p}$ norm.

2.1

The sequence $u = (1, 1 / 2, 1 / 3, \dots)$ lies in $c_{0}$ but not in $c_{00}$ , while its truncations $u^{(N)} \in c_{00}$ converge to $u$ in the supremum norm by step 1.1. Hence $c_{00}$ is not complete for that norm, and since $c_{0}$ is Banach by [L1], [L2] identifies the supremum-norm completion of $c_{00}$ with $c_{0}$ .

step 1.1L1L2

3.1

Because $ℓ^{p}$ is Banach by [L3], the uniqueness statement [L2] identifies the $ℓ^{p}$ -completion of $c_{00}$ with $ℓ^{p}$ .

step 1.2L2L3∎

Depends on

Used by

An incomplete normed subspace need not be closed Counterexample
Dictionary of the classical Lᵖ and ℓᵖ Banach spaces Example

Dependency tree · two levels

13 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)