LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 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.

A complete normed subspace is closed

Statement

Let $V$ be a normed space and let $W \subseteq V$ be a normed subspace. If $W$ is complete for its restricted norm, then $W$ is closed in $V$ .

Facts & Assumptions

Given: A normed space $V$ and a normed subspace $W \subseteq V$ .

[L1]

The normed-subspace metric is the ambient metric restricted to $W \times W$ , and the inclusion is an isometric embedding (Normed subspace).

[L2]

A complete subspace of any metric space is closed in the ambient space (A subspace of a complete metric space is complete iff it is closed, and a complete subspace of any metric space is closed).

Proof

technique · direct

1.1

By [L1], the metric on $W$ induced by the restricted norm is exactly the ambient norm metric restricted to $W$ .

2.1

The hypothesis that $W$ is complete for its restricted norm therefore says that $W$ is complete as a metric subspace of $V$ .

step 1.1given

3.1

Applying [L2] to that metric subspace shows that $W$ is closed in the ambient normed space $V$ .

step 2.1L2∎

Depends on

Used by

An incomplete normed subspace need not be closed Counterexample

Dependency tree · two levels

16 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)
Kyriakos Keremedis and Eliza Wajch, On densely complete metric spaces and extensions of uniformly continuous functions in ZF (standard reference, not scraped)