Approximation of analytic functions by rational functions in Banach spaces (Q1058114)

The author proves several resuls on the rational approximation of analytic functions over certain compact subsets of a Banach space E with the approximation property. An important tool in the proof of these results is a generalized version of the Shilov-Arens-Calderón- Waelbroeck theorem. As an application of this result a simple proof of the Oka-Weil theorem in E is given. This proof allows the athor to state a result on approximation of continuous functions by polynomials over totally disconnected polynomially convex compact subsets of E. A vector- valued version of the Oka-Weil theorem is proved, and the author shows that the vector Oka-Weil property for a Banach space implies the approximation property. The rational functions on an open subset U of E are introduced, and the normal envelope of rationality \(U_{{\mathcal R}}\) is constructed as a subset of E. These results are fundamental for the part where the author proves Oka-Weil type theorems concerning the approximation of continuous functions by rational functions. Some results relating rational, polynomial and holomorphic convexity are proved. When E has the bounded approximation property, the author shows that \(U_{{\mathcal R}}\) is a domain of holomorphy.