Weakly compact holomorphic mappings on Banach spaces

A holomorphic mapping f: E → F of complex Banach spaces is weakly compact if every x ∈ E has a neighbourhood V_x such that f(V_x) is a relatively weakly compact subset of F. Several characterizations of weakly 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 H(E) of holomorphic complex-valued functions on E, endowed with the bornological topology τ_δ, is reflexive.