The compact approximation property for spaces of holomorphic mappings on Fréchet spaces

Abstract

In this paper, the compact approximation property on Frechet spaces is characterized in terms of holomorphic mappings. We show that a Frechet space E has the compact approximation property if and only if every holomorphic mapping on a balanced open subset $$U\subset E$$ with values in a Frechet space can be approximated uniformly on compact subsets of U by compact holomorphic mappings. This extends the well-known linear characterization to the holomorphic setting. We also give characterizations of the compact approximation property in terms of bounded holomorphic mappings on Banach spaces.