A Hahn-Banach theorem for holomorphic mappings on localy convex spaces
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 \(G\supset E\).
scientific article; zbMATH DE number 3855763