I'm looking at differentiability, analyticity and power series functions between Banach spaces, I'm using Soo Bong Chae's book "Holomorphy and Calculus in Normed Spaces".
The book has the definition:
A power series from $E$ to $F$ about $\xi\in E$ is a formal series of the form $$\sum_{m=0}^{\infty}{A_m (x-\xi)^m}$$ where $A_m \in \mathcal{B}^m(E;F)$.
Here we have the idea of a Taylor series, we have the $m$ derivatives. On page 168, it is said who $A_m$ is, $$A_m = \frac{1}{m!}d^mf(\xi)$$ where $d^mf$ is the $m$-linear map, $d^mf: U\to \mathcal{B}^n(E,F)$.
So here is my doubt, the power series should be from $E$ to $F$? I'm probably misinterpreting something... should I consider $d^mf$ applied m times to $\xi$? But I didn't get that from the book.
In that case, I would also like to know if anyone has any book references or material that addresses complex analysis in Banach spaces.
Notation: $\mathcal{B}^n(E,F)$ is the space of all continuous multilinear mappings of $E\times\cdots E$ ($n$-times) into $F$.