Alphabeta Math
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⊆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∈W).

Its induced metric is the restriction of the ambient norm metric to W×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.

Depends on

Used by

Dependency tree · two levels

20 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