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holomorphically convex
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(Definition)
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Let $G \subset {\mathbb{C}}^n$ be a domain, or alternatively for a more general definition let $G$ be an $n$ dimensional complex analytic manifold. Further let ${\mathcal{O}}(G)$ stand for the set of holomorphic functions on $G$
Definition 1 Let $K \subset G$ be a compact set. We define the holomorphically convex hull of $K$ as \begin{equation*} \hat{K}_G := \{ z \in G \mid \lvert f(z) \rvert \leq \sup_{w \in K} \lvert f(w) \rvert \text{ for all } f \in {\mathcal{O}}(G) \} . \end{equation*}The domain $G$ is called holomorphically convex if for every $K \subset G$ compact in $G$ $\hat{K}_G$ is also compact in $G$ Sometimes this is just abbreviated as holomorph-convex.
Note that when $n=1$ any domain $G$ is holomorphically convex since when $n=1$ $\hat{K}_G = K$ for all compact $K \subset G$ Also note that this is the same as being a domain of holomorphy.
- 1
- Lars Hörmander. An Introduction to Complex Analysis in Several Variables, North-Holland Publishing Company, New York, New York, 1973.
- 2
- Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.
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"holomorphically convex" is owned by jirka.
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Cross-references: domain of holomorphy, compact, compact set, holomorphic functions, complex analytic manifold, domain
There is 1 reference to this entry.
This is version 5 of holomorphically convex, born on 2005-02-22, modified 2006-04-20.
Object id is 6798, canonical name is HolomorphicallyConvex.
Accessed 3438 times total.
Classification:
| AMS MSC: | 32E05 (Several complex variables and analytic spaces :: Holomorphic convexity :: Holomorphically convex complex spaces, reduction theory) |
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Pending Errata and Addenda
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