CorollaryStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-09

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Relative Hahn–Banach makes the canonical bidual map an isometry

Statement

Assume HB. For every real or complex normed space $X$ , the canonical scalar-linear map $J_{X} : X \to X^{* *}$ , given by $J_{X} (x) (f) = f (x)$ , satisfies $∥ J_{X} x ∥ = ∥ x ∥ (x \in X) .$ It preserves distances and is injective. Surjectivity is not claimed.

Facts & Assumptions

[F1]

Under HB, each nonzero $x$ has a functional $f$ with $∥ f ∥ = 1$ and $f (x) = ∥ x ∥$ (Relative dual norming, point separation, and recovery of the norm).

[F2]

Evaluation defines a scalar-linear $J_{X} : X \to X^{* *}$ with $∥ J_{X} x ∥ \leq ∥ x ∥$ (Evaluation defines a bounded scalar-linear map into the bidual).

Proof

Given: HB, a normed real or complex space $X$ , and the evaluation map $J_{X}$ .

1.1

By the evaluation construction, $J_{X}$ is scalar-linear and $∥ J_{X} x ∥ \leq ∥ x ∥$ for every $x$ . In particular $J_{X} 0 = 0$ and equality of the norms holds at zero.

givenF2

2.1

For $x \neq 0$ , HB norming gives $f \in X^{*}$ with $∥ f ∥ = 1$ and $f (x) = ∥ x ∥$ . The bidual norm is the supremum over the dual unit ball, which contains this $f$ , so $∥ J_{X} x ∥ \geq ∣ J_{X} x (f) ∣ = ∣ f (x) ∣ = ∥ x ∥$ . Combining with step 1.1 proves equality at every $x$ .

step 1.1F1F2

3.1

For $x, y \in X$ , linearity and the established equality give $∥ J_{X} x - J_{X} y ∥ = ∥ J_{X} (x - y) ∥ = ∥ x - y ∥$ . If $J_{X} x = J_{X} y$ , the left side is zero, so definiteness of the norm gives $x = y$ .

step 1.1step 2.1algebra∎

Source notes

Brezis §1.3, pp.8–9, first displayed isometry calculation; Teschl Theorem 4.20, pp.115–116.

Depends on

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Sources

Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, §§1.1–1.2 and §1.3 evaluation paragraph (standard reference, not scraped)
Gerald Teschl, Topics in Real and Functional Analysis, Theorems 4.13–4.20 and §5.1 (2018 university-hosted copy) (standard reference, not scraped)