I am posting a CW answer. Feel free to add other references.
Here are some references containing at least some basic facts about the construction from the question (or at least the fact that it yields a Banach space).
Books
- Aliprantis, Charalambos D.; Border, Kim C. (2006). Infinite dimensional analysis: A hitchhiker's guide (Third ed.). Berlin: Springer; page 553
- Helemskii: Lectures and exercises on functional analysis, AMS 2006; Proposition 1.1.7 and remark on page 127
- N. L. Carothers: A Short Course on Banach Space Theory, CUP, 2004; page 49
Online resources
- R. Shvydkov, Lectures on functional analysis, http://homepages.math.uic.edu/~shvydkoy/math539/LecturesonFA.pdf. (Section 1.3).
- Sum of Banach Spaces is complete
Searches
This construction is sometimes called the $l_p$-sum of Banach spaces $X_1,X_2,\dots$. If we know the name, this might help it finding at least some references if we search for this name with some reasonable additional keywords: