DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-28

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.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Holomorphic hulls and holomorphic convexity

Definition

Let $Ω \subseteq C^{m}$ be a domain and let $E \subseteq Ω$ . The holomorphic hull of $E$ in $Ω$ is

${\hat{E}}_{Ω} : = {a \in Ω : ∣ f (a) ∣ \leq \sup_{z \in E} ∣ f (z) ∣ for every f \in O (Ω)},$

where the supremum is taken in $[0, \infty]$ .

If $K ⋐ Ω$ is compact, then $Ω$ is holomorphically convex when ${\hat{K}}_{Ω} ⋐ Ω$ for every such $K$ .

Remarks

The extended supremum is deliberate. For an arbitrary set $E$ , some holomorphic functions may be unbounded on $E$ , and then the corresponding inequality in the definition is automatic.

The empty-set case is harmless: the constant function $1$ shows ${\hat{\emptyset}}_{Ω} = \emptyset$ , because $∣ 1 (a) ∣ \leq \sup_{\emptyset} ∣ 1 ∣ = 0$ fails at every point.

Depends on

Holomorphic functions on an open subset of $\mathbb{C}^m$

Used by

Dependency tree · two levels

10 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

Jiří Lebl, Tasty Bits of Several Complex Variables, §2.6 (standard reference, not scraped)
Harold P. Boas, Lecture Notes on Several Complex Variables, §3.2.3 (standard reference, not scraped)