CorollaryStatement: AI-adaptedProof: AI-adaptedPipeline-generatedverified 2026-09-24 (gpt-6-sol)

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Distance to an annihilator is the restriction norm

Statement

Assume the Axiom of Choice. Let $K = R$ or $C$ . If $M$ is a closed linear subspace of a normed $X$ and $f \in X^{*}$ , then $dist (f, M^{⊥}) = ∥ f ∣_{M} ∥ .$

Facts & Assumptions

Given: The Axiom of Choice, 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.

[A1]

AC is used through the restriction isometry [F1] in step 2.1.

[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 $M \leq X$ closed. Restriction $R : X^{*} \to M^{*}$ induces a linear isometric bijection $\tilde{R} : X^{*} / M^{⊥} ⟶ M^{*}, f + M^{⊥} ⟼ f ∣_{M} .$ Also $∥ R ∥ \leq 1$ ; its norm is $1$ when $M \neq {0}$ and $0$ when $M = {0}$ .

Proof

1.1givenalgebra

The quotient norm is $∥ f + M^{⊥} ∥ = \inf_{a \in M^{⊥}} ∥ f + a ∥ = dist (f, M^{⊥})$ , since $M^{⊥}$ is a linear subspace.

2.1F1step 1.1∎

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

Depends on

Used by

Banach closed-range theorem Theorem

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 2.58, (2.33), p.85 (standard reference, not scraped)