Differentiating term by term in a Banach space: how to justify it?

Question

After looking at this question, I am now wondering if the theorem proven in the first answer below can be generalized to a Banach space. See here for my attempt. But before doing that, I have the following problem:

NOTE: I use the notations in the first answer below that question.

I don't know how to justify why the series $\sum a_kf_k$ can be differentiated term by term, i.e. why $\partial^\alpha (\sum a_kf_k)=\sum a_k(\partial ^\alpha f_k) $, and why the convergence of $\sum a_k\partial ^\alpha f_k$ implies the existence of $\partial^\alpha (\sum a_kf_k)$

In Banach space, the multi-index notation does not mean anything, so I should write $\sum a_kD^nf_k$.

Answer 1

(Note: This is just a part of the answer, I am not even sure if the whole proof can be generalized to separable Banach spaces.)

I don't know how to justify why the series $\sum a_kf_k$ can be differentiated term by term, i.e. why $\partial^\alpha (\sum a_kf_k)=\sum a_k(\partial ^\alpha f_k) $, and why the convergence of $\sum a_k\partial ^\alpha f_k$ implies the existence of $\partial^\alpha (\sum a_kf_k)$

A possible way to solve this is by verifying that $\sum a_k D^nf_k$ is a Fréchet derivative of $\sum a_k D^{n-1} f_k$ (and then use induction). In order to show this the estimate $$ \| D^{n-1}f_k(x+h)- D^{n-1} f_k(x) - D^n f_k(x)h\| \leq \sup_{\|y\|\leq \|h\|} \frac12 \| D^{n+1} f_k(x+y)\| \|h\|^2 $$ can be useful (this estimate follows by using a Taylor series and estimating the remainder).