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Basic properties of the holomorphic hull

Statement

Let $Ω \subseteq C^{m}$ be a domain.

For every $E \subseteq Ω$ , one has $E \subseteq {\hat{E}}_{Ω}$ .
If $K \subseteq Ω$ is compact, then ${\hat{K}}_{Ω}$ is closed in $Ω$ and is bounded in each coordinate.
For every $E \subseteq Ω$ , one has ${\hat{{\hat{E}}_{Ω}}}_{Ω} = {\hat{E}}_{Ω} .$

Facts & Assumptions

Given: A domain $Ω \subseteq C^{m}$ , a subset $E \subseteq Ω$ , and a compact set $K \subseteq Ω$ .

[L1]

The holomorphic hull is defined by the pointwise inequalities $∣ f (a) ∣ \leq \sup_{E} ∣ f ∣$ for every holomorphic $f$ on $Ω$ (Holomorphic hulls and holomorphic convexity).

Proof

technique · direct

1.1L1given

If $a \in E$ and $f \in O (Ω)$ , then $∣ f (a) ∣ \leq \sup_{E} ∣ f ∣$ by definition of the supremum. Hence [L1] gives $a \in {\hat{E}}_{Ω}$ , so $E \subseteq {\hat{E}}_{Ω}$ .

1.2L1given

For compact $K$ , [L1] gives ${\hat{K}}_{Ω} = ⋂_{f \in O (Ω)} {a \in Ω : ∣ f (a) ∣ \leq \sup_{K} ∣ f ∣} .$ Each set in the intersection is closed in $Ω$ because $f$ is continuous, so ${\hat{K}}_{Ω}$ is closed in $Ω$ . The coordinate functions $z \mapsto z_{j}$ are holomorphic on $Ω$ , so [L1] also gives $∣ a_{j} ∣ \leq \sup_{z \in K} ∣ z_{j} ∣$ for every $a \in {\hat{K}}_{Ω}$ and every coordinate $j$ . Thus ${\hat{K}}_{Ω}$ is coordinate-bounded.

2.1L1step 1.1algebra∎

Step 1.1 applied to $E$ gives ${\hat{E}}_{Ω} \subseteq {\hat{{\hat{E}}_{Ω}}}_{Ω}$ . For the reverse inclusion, let $a \in {\hat{{\hat{E}}_{Ω}}}_{Ω}$ . Then [L1] gives $∣ f (a) ∣ \leq \sup_{{\hat{E}}_{Ω}} ∣ f ∣$ for every $f \in O (Ω)$ , while the definition of ${\hat{E}}_{Ω}$ itself gives $\sup_{{\hat{E}}_{Ω}} ∣ f ∣ \leq \sup_{E} ∣ f ∣$ . So $∣ f (a) ∣ \leq \sup_{E} ∣ f ∣$ for every holomorphic $f$ , and another use of [L1] shows $a \in {\hat{E}}_{Ω}$ .

Depends on

Holomorphic hulls and holomorphic convexity

Used by

Dependency tree · two levels

2 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)