Let $B$ be a complex Banach space and $H,I_n:B\to\mathbb{C}$ be analytic functions such that on a dense subset $$ \sum_{n=0}^\infty I_n = H. $$ Does it hold that $\sum_{n=0}^\infty I_n$ is an analytic function and hence $$ \sum_{n=0}^\infty I_n = H $$on $B$?
If it is of any help, the dense subset is given by $\{ I_n\not=0$ for finitely many $n\geq0\}$.
To rephrase the question, do I need some sort of convergence of the infinite sum or do I have a chance since the equality holds on a dense subset?