DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-09

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The finite gauge of an open convex neighbourhood of zero

Definition

Let $X$ be a real or complex normed space and let $U \subseteq X$ be open and convex with $0 \in U$ , using Convex sets and continuous real-hyperplane separation in a normed space. For $x \in X$ set $S_{x} = {t \in R : t > 0, x / t \in U}, p_{U} (x) = \inf S_{x} .$ The function $p_{U} : X \to R$ is the gauge of $U$ , with real nonnegative values and real positive scale parameters. Symmetry and boundedness of $U$ are not assumed.

This infimum is well-defined without HB or choice. Openness at zero gives one $r > 0$ with $B (0, r) \subseteq U$ . For any fixed $x$ and any $t > ∥ x ∥ / r$ , $∥ x / t ∥ < r$ , so $t \in S_{x}$ . Thus $S_{x}$ is nonempty, for example at $t = 1 + ∥ x ∥ / r$ , and is bounded below by zero. The real infimum property Every nonempty set bounded below has an infimum supplies a finite $p_{U} (x) \geq 0$ . These formulas define a unique value at every $x$ ; no family of choices is involved. At zero, $S_{0} = (0, \infty)$ , whose infimum is zero because it has members below every positive number.

The scale sets give the following direct calculations. For $U = B (0, 1)$ , one has $S_{x} = {t > 0 : t > ∥ x ∥}$ , so $p_{U} (x) = ∥ x ∥$ , including zero. For the open convex strip $U = {(s, v) \in R^{2} : ∣ s ∣ < 1}$ , the condition $(s, v) / t \in U$ is exactly $t > ∣ s ∣$ , so $p_{U} (s, v) = ∣ s ∣$ . This set contains $(0, v)$ for all real $v$ and is unbounded; its gauge vanishes along that whole line. Finally, $U = {0}$ in a nonzero normed space is convex but not a neighbourhood of zero: every positive-radius ball contains a nonzero multiple of any fixed nonzero vector. For $x \neq 0$ its scale set is empty since $x / t \neq 0$ for every $t > 0$ . Thus that set does not define a finite gauge on all of $X$ by this construction.

Source notes

Brezis Lemma 1.2 and (8), p.6; Teschl (5.1) and Lemma 5.1, pp.137–138.

Depends on

Used by

The open convex gauge is sublinear and recovers its set Lemma

Dependency tree · two levels

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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)