Weakly compact holomorphic mappings on Banach spaces (Q1084302)

A holomorphic mapping \(f: E\to F\) of complex Banach spaces is weakly compact if every \(x\in E\) has a neighbourhood \(V_ x\) such that \(f(V_ x)\) is a relatively weakly compact subset of F. Several characterizations of weakly compact holomorphic mappings are given which are analogous to classical characterizations of weakly compact linear mappings and the Davis-Figiel-Johnson-Pelczynski factorization theorem is extended to weakly compact holomorphic mappings. It is shown that the complex Banach space E has the property that every holomorphic mapping from E into an arbitrary Banach space is weakly compact if and only if the space \({\mathcal H}(E)\) of holomorphic complex-valued functions on E, endowed with the bornological topology \(\tau _{\delta}\), is reflexive.