I'm trying to orient myself in the study of sequence-spaces. I'm particularly interested in geometric analogies between $\mathbb R^n$ and infinite dimensional sequence-spaces.
In this context for example, the fact that the series $\sum_n 1/2^n$ converges, does it have a geometric interpretation? I understand that in the finite case, the sum $1 + 1/2 + \dotsc + 1/2^n$ is the inner product of $(1,1/2, \dotsc, 1/2^n)$ and $(1,1,\dotsc 1)$, but in infinite dimensions the vector $(1,1, \dotsc )$ has infinite length, so how can the analogy be constructed?
So i'm asking for suggestions about introductory books studying sequence-spaces from the geometric viewpoint.
A useful book is : Cotlar & Cignoli, An Introduction to Functional Analysis