I'm guessing following Theorem would be right but I can't conclude my proof nore find any reference proving it.
Let $V$ be a real subspace of a complex Banach space $E$, such that $E$ is the complexification of $V$, and $U$ an open neighborhood. Let $f_n:U\to\mathbb{C}$ be analytic such that the sum $\sum_{n\geq0} |f_n|$ converges locally uniformly on $V$ then there is an open neighborhood $U'$ of $V$ in $U$ such that $\sum_{n\geq0}f_n$ converges to an analytic function on $U'$.
My thoughts are that, since $f_n$ are analytic and I have uniform convergence on the real subspace the sum is a real analytic function on the real subspace and hence can be extended to an open neighborhood $U'$ of $V$. Then I try to show that this function is, maybe on a smaller $U'$, the limit of the sum. The way I try to show it is
- I take a point in $V$, since the sum is real analytic it has a Taylor expansion with positive convergence radius around that point.
- Using the absolute convergence I show that the Taylor coefficients of the sum are the sum of the Taylor coefficients of the $f_n$'s.
Now I would need to show that by this, the analytic continuation on the ball where the Taylor series converges is actually the sum but I can only show that if the sum converges then it is represented by this Taylor series.
Can one finish this proof or is there a counter example explaining my difficulty to show this?