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holomorphically convex
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(Definition)
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Let
 be a domain, or alternatively for a more general definition let  be an  dimensional complex analytic manifold. Further let
 stand for the set of holomorphic functions on  .
Definition 1 Let
 be a compact set. We define the holomorphically convex hull of  as
 for all  |
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The domain  is called holomorphically convex if for every
 compact in  ,  is also compact in  . Sometimes this is just abbreviated as holomorph-convex.
Note that when  , any domain  is holomorphically convex since when 
 for all compact
 . 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 2691 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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