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I have posted the following question (with minimal differences) on MSE some days ago, without receiving a satisfactory answer, so let me try here again.


Take the vector space of infinitely differentiable functions on $[0,1]$. The standard norm of $C^k([0,1])$ is just the $\ell^1$-norm of the vector $(\|f\|_\infty, \|f'\|_\infty,\ldots,\|f^{(k)}\|_\infty)$, but of course this idea cannot be further pursued to define a norm on $C^\infty([0,1])$.

However, what if one would consider the space $$ \mathcal S_p:=\{f\in C^\infty([0,1]):(\|f^{(n)}\|_\infty)_{n\in\mathbb N}\in \ell^p \} $$ for $p\in [1,\infty]$? These spaces are certainly small - in particular, for $p<\infty$ $\mathcal S_p$ contains neither $\exp$, nor $\sin$ and $\cos$ - but at least they do contain the polynomials and they seem to be Banach spaces - in fact even Banach lattice algebras. Do these spaces appear in applications, especially in PDEs? Has anybody ever studied their functional analytical properties and if this is not the case, what are these spaces' most obvious drawbacks?

(@JochenWengenroth has already pointed out in a comment on MSE that partition of unity would not hold in these spaces.)

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    $\begingroup$ I'm taking the liberty of adding a "banach algebras" tag -- similar looking algebras have been considered in the 1970s, I think, but I would need to consult either Dales or the book of Dales. Sometimes one imposes even faster decay of the norms of the derivatives (with respect to weights) to get something like quasi-analyticity: see Definition 1.2 of dx.doi.org/10.1016/0022-1236(73)90065-7 $\endgroup$ – Yemon Choi May 8 '16 at 23:02
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There is a host of such spaces, they run under the general name of Denjoy-Carleman ultradifferentiable spaces of Sobolev type of Beurling character (global norms). If you make the norm dependent on the choice of a compact subset, you get those of Roumieu type, which are Frechet spaces. See for example:

  • Andreas Kriegl, Peter W. Michor, Armin Rainer: The exponential law for spaces of test functions and diffeomorphism groups. Indagationes Mathematicae 27, 1 (2016), 225–265, pdf
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  • $\begingroup$ Thanks! The space ${\mathcal B}([0,1],\mathbb R)$ in your paper agrees with my $\mathcal S_\infty$, but I am (still?) missing the $\mathcal S_p$-spaces for finite $p$. Besides, I have a question: why do you stress the Fréchet-space-structure of such space, if ${\mathcal B}([0,1],\mathbb R)=\mathcal S_\infty$ is actually even a Banach space? $\endgroup$ – Delio Mugnolo May 9 '16 at 7:46
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    $\begingroup$ @Delio Mugnolo: If you refer to the $\mathcal B(U,F)$ in 3.1 on page 5 here, then you should observe first that there $U$ is assumed to be an open set in the domain space, and second that it is only assumed that each derivative separately is bounded on $U$, not that all the derivatives have a "uniform" bound. So the Michor et al. $\mathcal B(U,F)$ is far from being your $\mathcal S_\infty$. $\endgroup$ – TaQ May 9 '16 at 20:03
  • $\begingroup$ @TaQ Ah, thanks, good point. Whence the Fréchet space structure. $\endgroup$ – Delio Mugnolo May 9 '16 at 20:12
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    $\begingroup$ @Delio Mugnolo: In my situation the subspace $F$ with underlying set $S$ of $E=C^{\kern.3mm\infty}(I)$ has the "induced" trace topology from $E$ which can also be viewed as the initial one under the map $F\to G$ given by $x\mapsto\langle\kern1mm{\rm D}^{\,i}x:i\in\mathbb N_0\,\rangle$ when as $G$ one takes the Fréchet space $\prod\,\boldsymbol E$ which is the product space of the countable family $\boldsymbol E=\mathbb N_0\times\{\,C\,(I)\,\} \kern.8mm$. (cont.) $\endgroup$ – TaQ May 14 '16 at 14:39
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    $\begingroup$ (cont.) Your situation is otherwise similar except that one has the Banach(able) space $G=\ell\kern.6mm^p(\boldsymbol E)\kern.8mm$, and hence $F$ gets a finer topology than the trace from $E\kern.4mm$. $\endgroup$ – TaQ May 14 '16 at 14:42
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For the spaces you are looking for, you could consult the monograph "Sobolev spaces of infinite order and differential equations" by Ju. Dubinskij in which he considers a palette of such spaces depending on three parameters. He published prolifically on this theme---motivated by work on pde's of infinite order---see his entry in MathSciNet. There is a review of his book in the Bulletin by A. Kufner. Interestingly, in this general framework the proof of the non triviality of the spaces is an issue.

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  • $\begingroup$ Thanks, interesting reference. Unfortunately, I don't have the book at hand and Kufner's review does not clearly reveal whether any condition about the families $a_\alpha,p_\alpha,r_\alpha$ are imposed (in "my" case one would need constant families of values $1,p,\infty$, respectively. And as said, I do think that in the case of bounded domains non-triviality should be obvious, thanks to the very existence of polynomials of finite order. That said, I've never seen a PDE of infinite order, but I do know some investigations of the wave equation in $C^\infty$, whence my original question. $\endgroup$ – Delio Mugnolo May 9 '16 at 10:50
  • $\begingroup$ Constancy of these sequences just simplifies thing and I don't think that the domain being bounded helps that much but you should check this in the text which is available, at least as a torso, on Google Books. $\endgroup$ – goleta May 9 '16 at 12:05
  • $\begingroup$ Well, general finite-order polynomials are bounded functions only whenever restricted to bounded domains... $\endgroup$ – Delio Mugnolo May 9 '16 at 12:11
  • $\begingroup$ Yes. my comment was rather sloppy. I meant that in the general case considered by Dubinskij (i.e., where the sequence of orders of derivation are not the camonical ones) the question of non-triviality is, dare I say it, non trivial. He gives a necessary and sufficient condition for non-triviality in his monographh. $\endgroup$ – goleta May 9 '16 at 15:05

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