Holomorphic Functions Between Banach Spaces
More generally, given two Banach spaces X and Y over the complex numbers and an open set U in 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.
Read more about this topic: Infinite-dimensional Holomorphy
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