A Hahn-Banach theorem for holomorphic mappings on localy convex spaces
type: Item
instance of: scholarly article
title: A Hahn-Banach theorem for holomorphic mappings on localy convex spaces
zbMATH Open document ID: 0538.46033
DOI: 10.1007/BF01159181
author: Ralf Hollstein
published in: Mathematische Zeitschrift
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173544
review text: A locally convex space E is called \(\epsilon\)-space if for each absolutely convex, closed 0-neighbourhood U in E there is an absolutely convex, closed 0-neighbourhood V, \(V\subset U\), such that the canonical mapping \(K_{UV}:\tilde E_ V\to \tilde E_ U\) is \(\infty\)-factorable, i.e. there exist an \(L_{\infty}(\mu)\)-space F and continuous linear mappings \(A:\tilde E_ V\to F\) and \(B:F\to \tilde E_ U\!''\) such that \(I_ U{\mathbb{O}}K_{UV}=B{\mathbb{O}}A\) where \(I_ U:\tilde E_ U\to \tilde E_ U\!''\) denotes the evaluation mapping. In this note it is shown that a locally convex space E is an \(\epsilon\)-space if and only if for every locally convex space F and every locally convex space G containing E as a topological linear subspace every holomorphic mapping \(f:E\to F\) of uniformly bounded type has a holomorphic extension \(\tilde f:G\to F_ n\!''\) where \(F_ n\!''\) denotes the bidual of F equipped with the natural topology. This theorem gives various counterexamples and generalizes a result of \textit{R. M. Aron} and \textit{P. D. Berner} [Bull. Soc. Math. France 106, 3-24 (1978; Zbl 0378.46043)] concerning holomorphic extensions in Banach spaces. Furthermore the extension theorem of \textit{P. J. Boland} [Trans. Am. Math. Soc. 209, 275-281 (1975; Zbl 0317.46036)] can be improved for not necessarily nuclear spaces E in the following way: If E is an \(\epsilon\)-(DFS)-space, then every Fréchet-valued holomorphic function on E can be holomorphically extended to each locally convex space \(G\supset E\).
Mathematics Subject Classification ID: 46G20
Mathematics Subject Classification ID: 46A04
Mathematics Subject Classification ID: 46A22
zbMATH DE Number: 3855763
zbMATH Keywords: \(\epsilon\)-space
zbMATH Keywords: holomorphic mapping
zbMATH Keywords: holomorphic extensions in Banach spaces
zbMATH Keywords: Fréchet-valued holomorphic function
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3855763