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| In [[mathematics]], more precisely in [[complex analysis]], the '''holomorphically convex hull''' of a given [[compact set]] in the ''n''-[[dimension]]al [[complex number|complex]] space '''C'''<sup>''n''</sup> is defined as follows.
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| Let <math>G \subset {\mathbb{C}}^n</math> be a domain (an [[open set|open]] and [[connected set|connected]] [[Set (mathematics)|set]]), or alternatively for a more general definition, let <math>G</math> be an <math>n</math> dimensional [[complex analytic manifold]]. Further let <math>{\mathcal{O}}(G)</math> stand for the set of [[holomorphic function]]s on <math>G.</math> For a compact set <math>K \subset G</math>, the '''holomorphically convex hull''' of <math>K</math> is
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| :<math> \hat{K}_G := \{ z \in G \big| \left| f(z) \right| \leq \sup_{w \in K} \left| f(w) \right| \mbox{ for all } f \in {\mathcal{O}}(G) \} .</math>
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| (One obtains a narrower concept of '''polynomially convex hull''' by requiring in the above definition that ''f'' be a [[polynomial]].)
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| The domain <math>G</math> is called '''holomorphically convex''' if for every <math>K \subset G</math> compact in <math>G</math>, <math>\hat{K}_G</math> is also compact in <math>G</math>. Sometimes this is just abbreviated as ''holomorph-convex''.
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| When <math>n=1</math>, any domain <math>G</math> is holomorphically convex since then <math>\hat{K}_G</math> is the union of <math>K</math> with the relatively compact components of <math>G \setminus K \subset G</math>. Also note that being holomorphically convex is the same as being a [[domain of holomorphy]] (The Cartan–Thullen theorem). These concepts are more important in the case ''n'' > 1 of [[several complex variables]].
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| ==See also==
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| * [[Stein manifold]]
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| * [[Pseudoconvexity]]
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| == References ==
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| * [[Lars Hörmander]]. ''An Introduction to Complex Analysis in Several Variables'', North-Holland Publishing Company, New York, New York, 1973.
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| * Steven G. Krantz. ''Function Theory of Several Complex Variables'', AMS Chelsea Publishing, Providence, Rhode Island, 1992.
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| {{PlanetMath attribution|id=6798|title=Holomorphically convex}}
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| [[Category:Several complex variables]]
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