Linearization of holomorphic mappings on locally convex spaces
Throughout this paper the letters \(E\) and \(F\) represent locally convex spaces, always assumed complex and Hausdorff, and the letter \(U\) represents a nonvoid open subset of \(E\). \({\mathcal L} (E; F)\) denotes the vector space of all continuous linear mappings form \(E\) into \(F\), whereas \({\mathcal H} (U; F)\) denotes the vector space of all holomorphic mappings from \(U\) into \(F\). \textit{P. Mazet} [``Analytic sets in locally convex spaces'', North Holland Math. Stud. 89 (1984; Zbl 0588.46032)] proved the existence of a complete locally convex space \(G(U)\) and a mapping \(\delta_U\in {\mathcal H} (U; G(U))\) with the following universal property: For each complete locally convex space \(F\) and each mapping \(f\in {\mathcal H} (U; F)\), there is a unique mapping \(T_f\in {\mathcal L} (G(u); F)\) such that \(T_f\circ \delta_U= f\). To prove this result Mazet introduced the notion of cotopological space and exploited the duality between cotopological spaces and locally convex spaces. In this paper we present a different proof of the Mazet linearization theorem, based on a result on inductive limits of Banach spaces. The rest of the paper is devoted to the study of some aspects of the interplay between the spaces \({\mathcal H} (U; F)\) and \({\mathcal L} (G (U); F)\), with applications to the study of holomorphically barrelled domains, holomorphically Mackey domains, holomorphic continuation, analytic sets and holomorphic convexity.
scientific article; zbMATH DE number 810300