Editing Infinite-dimensional holomorphy (section)

==Holomorphic functions between Banach spaces==
More generally, given two complex [[Banach space]]s ''X'' and ''Y'' and an open set ''U'' ⊂ ''X'', ''f'' : ''U'' → ''Y'' is called '''holomorphic''' if the [[Fréchet derivative]] of ''f'' exists at every point in ''U''. One can show that, in this more general context, it is still true that a holomorphic function is analytic, that is, it can be locally expanded in a [[power series]]. It is no longer true however that if a function is defined and holomorphic in a ball, its power series around the center of the ball is convergent in the entire ball; for example, there exist holomorphic functions defined on the entire space which have a finite radius of convergence.{{ref|Harris}}