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.

The quotient vector space (X/M), its cosets, and the quotient map (q:X\to X/M)

Definition

Let X be a normed space and let M≤X be a linear subspace. The underlying quotient vector space is the one already defined in The quotient vector space V/W and its canonical projection, with cosets written

x+M:={x+m:m∈M}.

On this page the canonical projection is written

q:X→X/M,q(x):=x+M.

By Coset equality, well-defined quotient operations, and the canonical projection with kernel W, q is a surjective linear map and ker⁡q=M.

Remarks

  • The quotient is algebraic at this stage; its norm is introduced next.
  • Different representatives of the same coset differ by an element of M.

Depends on

Used by

Dependency tree · two levels

5 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