If we are given countably many Banach spaces $X\_1, X\_2, \ldots, X\_k, \ldots$, then we can norm their Cartesian product by


$$
\left\| (x\_k)\_{k \in \mathbb N} \right\|\_p := \left( \sum\_k \| x\_k \|\_k^p \right)^{1/p},
$$


or the analogous construction for the case $p = \infty$. Trivially, the elements where this is $< \infty$ form a Banach space.


My question: Is there a reference where this construction is explained (and ideally, where the verifications are done)?



Reference on the countable product of Banach spaces
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))


