The regularity of spaces of germs of Fréchet-valued bounded holomorphic functions
type: Item
instance of: scholarly article
title: The regularity of spaces of germs of Fréchet-valued bounded holomorphic functions
zbMATH Open document ID: 0935.46034
author: Nguyen Ha Thanh
published in: Acta Mathematica Vietnamica
publication date: 2000-04-24T00:00:00Z
review text: Let \(E, F\) be two Fréchet spaces and let \(K\) be a compact set in \(E\). Then the space of germs of \(F\)-valued bounded holomorphic functions on \(K\) is denoted by \(H_{\infty}(K, F)\). Here the author shows that \(F\) has the property \((DN)\) if and only if \(H_{\infty}(K, F)\) is regular for every compact set in any Fréchet space \(E\) with the property \((\Omega)\). Moreover the Fréchet space \(E\) has the property \((\Omega)\) if and only if \(H_{\infty}(K, F)\) is regular for every compact set in any Fréchet space \(F\) with the property \((DN)\). As an application it is shown that for nuclear Fréchet spaces \(E\) and \(F\) such that \(E\in (\Omega)\) and \(F\in\overline{(DN)}\), the space \(H(K, F)\) of germs of \(F\)-valued holomorphic functions on \(K\) is regular and complete.
Mathematics Subject Classification ID: 46E50
Mathematics Subject Classification ID: 46G20
Mathematics Subject Classification ID: 32A30
Mathematics Subject Classification ID: 46A63
Mathematics Subject Classification ID: 46A11
zbMATH DE Number: 1347401
zbMATH Keywords: germs of \(F\)-valued holomorphic functions
zbMATH Keywords: property \((DN)\)
zbMATH Keywords: nuclear Fréchet spaces
reviewed by: Jafar Zafarani
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1347401