4
$\begingroup$

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)?

Improve this question
$\endgroup$
3
  • 1
    $\begingroup$ I am not sure how widespread is this name, but this is sometimes called the $l_p$-sum of spaces $X_1,X_2,\dots$. This name is used for example in Helemskii's book. Lectures and Exercises on Functional Analysis. (Although only the sequences such that the above sue is finite are included. I guess that is what you meant.) Knowing the name might help you when looking for references: Google, Google Books. $\endgroup$
    – Martin Sleziak
    Commented Aug 31, 2016 at 11:17
  • 1
    $\begingroup$ math.stackexchange.com/questions/1620071/… $\endgroup$
    – Tomasz Kania
    Commented Aug 31, 2016 at 11:18
  • 1
    $\begingroup$ @TomekKania I'm not sure whether stackexchange is appropriate for citation in a research paper. $\endgroup$
    – Cloudscape
    Commented Aug 31, 2016 at 11:20

1 Answer 1

Reset to default
6
$\begingroup$

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

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:

Improve this answer
$\endgroup$
3
  • $\begingroup$ I have posted this answer based on the OP request in a comment, where they asked me to expand the comment to an answer. (The comment has been deleted in the meantime.) If the answer of this form is unsuitable for MO, let me know and I can delete it. (Or the whole question can be migrated to math.SE - I guess answer like this is acceptable there.) $\endgroup$
    – Martin Sleziak
    Commented Aug 31, 2016 at 12:29
  • $\begingroup$ Thanks for the effort! My goal was to be able to accept the answer, so that mathematicians who are in "answer mode" won't click on this question any more. $\endgroup$
    – Cloudscape
    Commented Aug 31, 2016 at 14:39
  • 1
    $\begingroup$ I should also say that from the last three books, only the Hemelskii book both mentions the case $p = \infty$ and simultaneously the fact that they are Banach spaces (the latter is omitted in Carothers' book). $\endgroup$
    – Cloudscape
    Commented Sep 2, 2016 at 10:34

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .