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

Used by

Dependency tree · two levels

14 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

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