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 a closed subspace via unit annihilators

Statement

Let $K = R$ or $C$ . For a closed linear subspace $M$ of a normed $X$ and $x \in X$ , $dist (x, M) = \sup {∣ f (x) ∣ : f \in M^{⊥}, ∥ f ∥ \leq 1} .$

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 quotient is its annihilator, with its stated hypotheses: Let $K = R$ or $C$ . Let $X$ be normed and $M \leq X$ closed. With quotient norm $∥ x + M ∥ = \inf_{m \in M} ∥ x + m ∥$ and $q (x) = x + M$ , the map $Q : (X / M)^{*} ⟶ M^{⊥}, Q h = h \circ q$ is a linear isometric bijection.

[F2]

From The canonical bidual map is an isometry, with its stated hypotheses: Let $K = R$ or $C$ . For every normed $X$ , the map $J_{X} : X \to X^{* *}$ is linear and $∥ J_{X} x ∥ = ∥ x ∥$ for every $x \in X$ . In particular it is injective.

Proof

1.1

In the normed quotient $Z = X / M$ , the bidual isometry gives $∥ x + M ∥ = \sup_{h \in Z^{*}, ∥ h ∥ \leq 1} ∣ h (x + M) ∣$ . The quotient norm on the left is $dist (x, M)$ .

2.1

The quotient-dual isometry sends its unit ball onto the unit ball of $M^{⊥}$ , with $f (x) = h (x + M)$ . Substitution proves the formula. The unit balls always contain zero; if $x \in M$ or $M = X$ both sides are zero, and $M = 0$ recovers the dual norm formula.

F1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

Bühler–Salamon, Functional Analysis, Corollary 2.69, p.88 (standard reference, not scraped)