A holomorphic mapping
f : E → F of complex Banach spaces is weakly compact if every x ∈ E has a
neighbourhood Vx such that f(Vx) 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-Pełczynski 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 τδ, is
reflexive.
