CounterexampleConstruction: 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.

An incomplete normed subspace need not be closed

Statement refuted

Every incomplete normed subspace of a normed space is closed.

Facts & Assumptions

Given: The supremum-norm inclusion $c_{00} \subset c_{0}$ .

[L1]

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

[L2]

The space $c_{00}$ is incomplete and dense in $c_{0}$ for the supremum norm (The finitely supported sequences form an incomplete normed space with different standard completions).

[L3]

A complete normed subspace is closed (A complete normed subspace is closed).

Counterexample

technique · direct

1.1

By [L2], the subspace $c_{00}$ is incomplete.

1.2

By [L2], the same subspace is dense in $c_{0}$ , so if it were closed then it would equal all of $c_{0}$ . But $c_{00} \neq c_{0}$ , since for example $(1, 1 / 2, 1 / 3, \dots) \in c_{0} ∖ c_{00}$ .

L2algebra

2.1

Therefore $c_{00}$ is an incomplete normed subspace that is not closed, refuting the statement. The ambient space is Banach by [L1], and [L3] explains why no contradiction occurs: [L3] goes in the opposite direction.

step 1.1step 1.2L1L3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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)