In this paper, the compact approximation property on Fréchet spaces is characterized in terms of holomorphic mappings. We show that a Fréchet space E has the compact approximation property if and only if every holomorphic mapping on a balanced open subset with values in a Fréchet 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.
Çalışkan, E., Rueda, P. The compact approximation property for spaces of holomorphic mappings on Fréchet spaces. Rev Mat Complut 34, 185–201 (2021). https://doi.org/10.1007/s13163-019-00344-9
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