Holomorphic and Subharmonic Convexity

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


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.


Open Subset Compact Subset Extension Point Compact Closure Holomorphic Convex 
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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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