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Since decades, mathematicians are studying function spaces, discovering new structures more and more adapted for a general theory of functional analysis.

In that works, sequence spaces are generally seen as a very simple example of "function" spaces. They become interesting in approximation theory (for instance to obtain some interesting characterization of Besov spaces using wavelet theory), being isomorphic to some important function spaces.

However, I was unable to find a good reference discussing important questions of functional analysis for the special case of sequence spaces, and discussing the particularity of this "simple" case, actually the simpler in infinite dimension.

I precise a bit my request with the kind of topics I am interested in.

  • Definition of interesting sequence spaces: $\mathcal{l}_p$ and weighted-$\mathcal{l}_p$ spaces, the nuclear space $s$ of rapidly decreasing sequences and its dual, the space of slowly increasing functions, etc. Embeddings between such spaces.
  • Theory of operators (infinite matrices) from $s$ to $s'$, for which Schwartz' kernel theorem applies. General study of the inversion of such operators.
  • Measure theory on $s'$, where the Minlos-Bochner theorem applies. What can we measure on sequences? Then, application to a general theory of random sequences.

Do you know a book that deals with this kind of problems, in the most complete possible manner?

NB. I know that some mathematicians were studying the question of the nuclearity of sequences spaces in the 60's and 70's but I didn't find any systematic treatment focused on sequence spaces.

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    $\begingroup$ The first volume of "Classical Banach spaces" by Lindenstrauss and Tzafriri has lots of material on Banach sequence spaces. Some of the most important work on locally convex sequence spaces was instigated by Köthe (Stufenräume and gestufte Räume) and continued by Grothendieck, for which you could consult the former's treatise and the latter's dissertation. Another source is the theory of summability where various sequence spaces are important for the funcional analytic treatment (Garling and students). $\endgroup$
    – barcelos
    Commented Apr 25, 2014 at 13:14
  • $\begingroup$ Thanks a lot for these sources. By any chance, do you have any idea if the works of Köthe (and also the ones of Pietsch) can be found in english? $\endgroup$
    – Goulifet
    Commented Apr 25, 2014 at 13:37
  • $\begingroup$ The first volume appeared originally in german but an english version followed a few years later. The second volume was published in english. There is also an english version of Pietsch' monograph on nuclear spaces. (I would give you more precise references but, unfortunately, I have no access to a good library at the moment, nor to mathscinet). $\endgroup$
    – barcelos
    Commented Apr 25, 2014 at 15:04
  • $\begingroup$ Just to make sure: Schwartz space of rapidly decaying function is isomorphic to the space of sequences little "s". $\endgroup$
    – Abdelmalek Abdesselam
    Commented Apr 28, 2014 at 17:54
  • $\begingroup$ Yes it is, but this is not true in general for subspaces, like spaces of $l_p$-sequences for instance. Moreover, questions of measure theory can be asked in both context: for the Schwartz space of rapidly decaying functions and the space of rapidly decaying sequences. My concern is that these questions are similar, but "simpler" in some sense for sequences (for which we avoid for instance questions of regularity). That's why I am interested in sequence spaces. $\endgroup$
    – Goulifet
    Commented Apr 28, 2014 at 19:06

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