Alphabeta Math
CorollaryStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07
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.

Distance to an annihilator is the restriction norm

Statement

Let K=R or C. If M is a closed linear subspace of a normed X and f∈X∗, then dist⁡(f,M⊥)=∥f∣M∥.

Facts & Assumptions

Given: The spaces, maps, scalar field, and hypotheses in the statement above. All duals consist of linear functionals over the ambient field; evaluation has no conjugation.

[F1]

From The dual of a closed subspace is a dual quotient, with its stated hypotheses: Let K=R or C. Let X be normed and M≤X closed. Restriction R:X∗→M∗ induces a linear isometric bijection R~:X∗/M⊥⟶M∗,f+M⊥⟼f∣M. Also ∥R∥≤1; its norm is 1 when M≠{0} and 0 when M={0}.

Proof

1.1

The quotient norm is ∥f+M⊥∥=inf⁡a∈M⊥∥f+a∥=dist⁡(f,M⊥), since M⊥ is a linear subspace.

givenalgebra
2.1

The restriction isometry identifies this quotient norm with ∥f∣M∥. For f=0 both sides vanish; M=0 gives zero and M=X gives ∥f∥.

F1step 1.1∎

Depends on

Used by

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Sources