Is Banach space a correct context to study sequences and series? Recently, I have reviewed elementary analysis and I realized that every theorem in the text(Rudin-PMA) about series can be generalized to Banach space.
Here is an example.
Below is the theorem stated in the text:

Let $\sum a_n,  \sum b_n$ be convergent series in $\mathbb{C}$.
If $\sum a_n$ is absolutely convergent, then $\sum \sum_{k=0}^n a_k b_{n-k} = \sum a_n \sum b_n$.

I realized that this theorem can be generalized in the context of Banach space. That is,

Let $(V,\| \cdot \|)$ be a Banach space over $\mathbb{K}$
Let $\sum v_n$ be a convergent series in $V$
Let $\sum c_n$ be a convergent series in $\mathbb{K}$.
If one of those series is absolutely convergent, then $\sum \sum_{k=0}^n c_k v_{n-k} = \sum c_n \sum v_n$.

Just like the example, i found that ratio test, comparison test, Drichlet test etc. can be generalized into Banach space.
Is Banach space the right generalization to study series or is there another well-known context to study series?
 A: You observe correctly that many results about series of real or complex numbers generalize to Banach spaces. Since $\mathbb R$ and $\mathbb C$ are examples of Banach spaces it should not be surprising that some (or even a lot) of what is taught about real or complex analysis generalizes to general Banach spaces. Moreover, in many cases the true nature of a particular result may be obscured by irrelevant peculiarities of the real or complex number systems and it thus becomes illuminated when considered as a result of Banach spaces. Having said that, to say that Banach spaces is the correct context to study series would be to miss the mark by quite a bit. Series is just one aspect of analysis. Banach spaces have an extremely rich geometry and series are more of a tool. An important tool. Banach spaces were not designed to study series. In fact, I'm not even sure what the theory of series might look like. What would the important questions in such a theory be? If you can formulate such an objective better, then a suitable context can be found. 
