Reference on the countable product of Banach spaces 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)?
 A: 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.

*Megginson, Robert E. (1998). An Introduction to Banach Space Theory. Graduate Texts in Mathematics. 193. New York: Springer; Exercise 5.1.

*A. Ya. 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 (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:


*

*"lp sum" "Banach spaces" (Google Books) 

*"lp direct sum" "Banach spaces" (Google Books)  

*"lp sum" "normed spaces" (Google Books) 

