Timeline for Reference for a General Theory of Sequences?
Current License: CC BY-SA 3.0
10 events
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| May 2, 2014 at 19:56 | comment | added | Abdelmalek Abdesselam | so then what kind of applications and results do you have in mind? I mentioned the case of the Bochner-Minlos Theorem because you explicitly brought it up as your third bullet point in your question. | |
| May 2, 2014 at 16:14 | comment | added | Goulifet | I agree with you: "simpler" was inappropriate. But I was not thinking about the Minlos theorem. For functions, you have both smoothness and decay, that give you roughly speaking two parameters: $s$ and $p$, for instance when you define Sobolev spaces. For sequences, only the decay with let say $l_p$ spaces. Here is what I called "simpler". | |
| May 2, 2014 at 16:11 | comment | added | Goulifet | What I wanted to say was this. First, $\mathcal{S} \subset L_p \subset \mathcal{S}'$. This is also true for sequences. The isomorphism you mention, let us call him $J$, is between $s$ and $\mathcal{S}$ or between $s'$ and $\mathcal{S}'$. But you don't have $L_p = J(l_p)$. A decay property for sequences is equivalent with both a decay and smoothness property for functions. | |
| May 2, 2014 at 13:37 | comment | added | Abdelmalek Abdesselam | @Goulifet: I don't understand what you say about subspaces. $l_p$ is not a subspace of $s$, it's the other way around. The measure theory, e.g., Bochner-Minlos theorem is not "simpler" on $s'$ than on $S'$: it is EXACTLY THE SAME as per the topological vector space isomorphism I mentioned. Regularity is not avoided it is hidden in the decay of the sequences realizing this isomorphism, i.e., the $L^2$ inner products with Hermite functions aka the eigenvectors of the harmonic oscillator. | |
| Apr 28, 2014 at 19:06 | comment | added | Goulifet | 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. | |
| Apr 28, 2014 at 17:54 | comment | added | Abdelmalek Abdesselam | Just to make sure: Schwartz space of rapidly decaying function is isomorphic to the space of sequences little "s". | |
| Apr 25, 2014 at 15:04 | comment | added | barcelos | 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). | |
| Apr 25, 2014 at 13:37 | comment | added | Goulifet | 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? | |
| Apr 25, 2014 at 13:14 | comment | added | barcelos | 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). | |
| Apr 25, 2014 at 12:12 | history | asked | Goulifet | CC BY-SA 3.0 |