If I define a function $f$ so that
$$f(z)=\sum_{n=1}^{\infty}c_ng_n(z)$$
Why if the series converge uniformly in some disc centered in $b$ then $f$ is continous in $b$? (I'm considering $\mathbb{C}$ as the Banach space if it helps) Thanks in advance!!
Let $$f_m(z) = \sum_{n=1}^m c_ng_n(z).$$ Since all the $g_n$ are continuous at $b$ (otherwise the result fails), all the $f_m$ are continuous at $b$. By assumption, the sequence $(f_m)$ converges uniformly in a neighborhood of $b$. Recall that uniform limits of continuous functions are continuous. Thus, the uniform limit of the $(f_m)$ is continuous at $b$, so $f = \lim_{ m \rightarrow \infty } f_m$ is continuous at $b$.