DefinitionDefinition: Literature-sourcedProof: Not applicablejudge 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.

Completion of a normed space

Definition

Let $X$ be a normed space. A completion of $X$ is a pair $(\hat{X}, ι)$ such that

$\hat{X}$ is a Banach space (Banach space);
$ι : X \to \hat{X}$ is a linear isometry (Linear isometries and isometric isomorphisms);
$ι [X]$ is dense in $\hat{X}$ (Interior, closure, boundary, limit point, isolated point and dense subset of a metric space).

Equivalently, a completion of a normed space is a completion of its norm metric (A completion of a metric space: a complete metric space together with an isometric embedding onto a dense subspace) together with compatible linear structure on the complete space.

Remarks

The embedding is part of the data, exactly as for metric completions.
Uniqueness means uniqueness up to a linear isometric isomorphism commuting with the dense embeddings.

Depends on

Dependency tree · 2 levels

Sources