Holomorphy in convergence spaces

Abstract

We introduce a new approach to infinite dimensional holomorphy. Cast in the setting of closed-embedded linear convergence spaces and based on a categorical definition of derivative, our theory applies beyond the traditional open domains. It reaches certain domains with empty interior (that arise naturally in Fréchet spaces) and gives a fully fledged differential calculus. On open domains our approach provides a new characterization of holomorphic maps. Thus earlier theories become expanded as well as strengthened.