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.

Normed subspace

Definition

Let $V$ be a normed space and let $W \subseteq V$ be a linear subspace in the sense of Linear subspace of a vector space. The normed subspace $W$ is the vector space $W$ equipped with the restricted norm $∥ w ∥_{W} : = ∥ w ∥_{V} (w \in W) .$

Its induced metric is the restriction of the ambient norm metric to $W \times W$ , so the inclusion $W ↪ V$ is an isometric embedding in the sense of Isometry, isometric embedding, and the subspace metric on a subset.

Remarks

No new notation is forced: once the ambient norm is fixed, the restricted norm is usually written with the same symbol.
Completeness and closedness for a normed subspace are therefore comparison statements between the restricted metric and the ambient one, not new definitions.

Normed subspace

Definition

Remarks

Depends on

Dependency tree · 2 levels

Sources