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](https://books.google.com/books?id=4hIq6ExH7NoC&pg=PA553).
 * Megginson, Robert E. (1998). An Introduction to Banach Space Theory. Graduate Texts in Mathematics. 193. New York: Springer; [Exercise 5.1](https://books.google.sk/books?id=fD-GeCsqoqkC&pg=PA439).
 * A. Ya. Helemskii: *Lectures and exercises on functional analysis*, AMS 2006;  [Proposition 1.1.7](https://books.google.com/books?id=wjzZCLzx6hUC&pg=PA63) and remark on [page 127](https://books.google.com/books?id=wjzZCLzx6hUC&pg=PA127).
 * N. L. Carothers: *A Short Course on Banach Space Theory*,  CUP, 2004; [page 49](https://books.google.com/books?id=pec4r3EsIiMC&pg=PA49).
 
**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](https://math.stackexchange.com/questions/1620071/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:

 * ["lp sum" "Banach spaces"](https://www.google.com/search?q=%22lp+sum%22+%22Banach+spaces%22) ([Google Books](https://www.google.com/search?q=%22lp+sum%22+%22Banach+spaces%22&tbm=bks)) 
 * ["lp direct sum" "Banach spaces"](https://www.google.com/search?q=%22lp+direct+sum%22+%22Banach+spaces%22) ([Google Books](https://www.google.com/search?q=%22lp+direct+sum%22+%22Banach+spaces%22&tbm=bks))  
 * ["lp sum" "normed spaces"](https://www.google.com/search?q=%22lp+sum%22+%22normed+spaces%22) ([Google Books](https://www.google.com/search?q=%22lp+sum%22+%22normed+spaces%22&tbm=bks))