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Holomorphic and Subharmonic Convexity

  • Charles E. Rickart
Part of the Universitext book series (UTX)

Abstract

It will be assumed throughout this section that [Σ, a] is a natural system. Also let {Σ, ℱ} be an a-presheaf of continuous functions over Σ. (See Definition 22.1). If Ω is any subset of Σ and K ⊂⊂ Ω then the ℱ-convex hull of K in Ω is the set
$$ K_{\Omega }^{\Im } = \left\{ {\sigma ' \in \Omega :\left| {f(\sigma ')} \right| \leqslant {{\left| f \right|}_K},\,f \in {\Im_{\Omega }}} \right\} $$
Since elements of ℱ are continuous the set \( K_{\Omega }^{\Im } \) is always relatively closed in Ω. Also, since |a| f ⊆ |ℱ| it follows that \( K_{\Omega }^{\Im }\underline \subset \hat{K} \cap \Omega \), so the closure of \( K_{\Omega }^{\Im } \) is compact.

Keywords

Open Subset Compact Subset Extension Point Compact Closure Holomorphic Convex 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York Inc. 1979

Authors and Affiliations

  • Charles E. Rickart
    • 1
  1. 1.Department of MathematicsYale UniversityNew HavenUSA

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