PlanetMath: Kontinuitätssatz

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Kontinuitätssatz

(Theorem)

Theorem 1 $G \subset {\mathbb{C}}^n$ is pseudoconvex if and only if for any family of closed analytic discs $\{ d_{\alpha} \}_{\alpha \in I}$ in with $\cup_{\alpha \in I} \partial d_\alpha$ being a relatively compact set in then $\cup_{\alpha \in I} d_\alpha$ is also a relatively compact set in .

This is the analogue of one of the definitions of a convex set. Just replace pseudoconvex with convex and closed analytic discs with closed line segments.

Bibliography

1: M. Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. Real Submanifolds in Complex Space and Their Mappings, Princeton University Press, Princeton, New Jersey, 1999.
2: Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.

"Kontinuitätssatz" is owned by jirka.

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Other names:

Hartogs Kontinuitätssatz, Kontinuitatssatz

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Cross-references: closed line segments, convex set, definitions, relatively compact, closed analytic discs, pseudoconvex
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This is version 2 of Kontinuitätssatz, born on 2006-03-30, modified 2006-03-30.
Object id is 7787, canonical name is Kontinuitatssatz.
Accessed 1779 times total.

Classification:

AMS MSC:

32T05 (Several complex variables and analytic spaces :: Pseudoconvex domains :: Domains of holomorphy)

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