A general approach to infinite-dimensional holomorphy
The purpose of the paper is to present a frame for a theory of holomorphic functions between locally convex spaces and bornological vector spaces respectively. Using continuous convergence [see \textit{E. Binz}, Lect. Notes Math. 469 (1975; Zbl 0306.54003)] instead of topologies on spaces of holomorphic functions a very general theory, which is remarkably simple to handle, is developed. A very general exponential law H(U\(\times V,G)\cong H(U,H(V,G))\) is proved. It yields well-known results concerning hypoanalytic functions and Gâteaux holomorphic functions as special cases [see \textit{S. Dineen}, Complex analysis in locally convex spaces (1981; Zbl 0484.46044)]. Furthermore, continuous convergence has very simple completeness properties on spaces of holomorphic functions. This shows clearly that continuous convergence is a very natural structure on these spaces. Further a new locally convex topology is introduced on spaces of holomorphic functions. This topology (the finest locally convex topology coarser than local uniform convergence) is compared with other locally convex topologies often used. Using an integral, which behaves nicely together with continuous convergence, power series expansions of holomorphic functions are derived. These are used for the proof of the exponential law.
holomorphic functions between locally convex spaces and
holomorphic functions between locally convex spaces and bornological vector spaces
scientific article; zbMATH DE number 3927660
scientific article