Holomorphic Approximation in Banach Spaces: A Survey

We give a survey about the Runge approximation problem for a holomorphic function defined on the unit ball of a complex Banach space. More precisely, we ask whether such a holomorphic function can be uniformly approximated on smaller balls by functions that are holomorphic on the entire space. This turns out to be a subtle (open) question, whose (partial) resolution in the past 15 years played a central role in deeper investigations in complex analysis in Banach spaces.