[Sous-espaces fermés universels dans des espaces de fonctions indéfiniment dérivables]
On exhibe les premiers exemples d’espaces de Fréchet contenant un sous-espace fermé de dimension infinie de séries universelles, mais ne contenant aucune série universelle restreinte. Pour cela, on considère les espaces de Fréchet classiques de fonctions indéfiniment dérivables qui n’admettent pas de norme continue. On établit alors des résultats plus généraux pour des suites d’opérateurs qui agissent sur des espaces de Fréchet avec ou sans norme continue. Enfin, on caractérise complètement l’existence de sous-espaces fermés de séries universelles dans l’espace de Fréchet .
We exhibit the first examples of Fréchet spaces which contain a closed infinite dimensional subspace of universal series, but no restricted universal series. We consider classical Fréchet spaces of infinitely differentiable functions which do not admit a continuous norm. Furthermore, this leads us to establish some more general results for sequences of operators acting on Fréchet spaces with or without a continuous norm. Additionally, we give a characterization of the existence of a closed subspace of universal series in the Fréchet space
Keywords: infinitely differentiable real functions, spaceability, universality, universal series, Taylor series
Mot clés : fonctions indéfiniment dérivables, sous-espaces fermés universels, universalité, séries universelles, séries de Taylor.
@article{AIF_2014__64_1_297_0,
author = {Charpentier, St\'ephane and Menet, Quentin and Mouze, Augustin},
title = {Closed universal subspaces of spaces of~infinitely differentiable functions},
journal = {Annales de l'Institut Fourier},
pages = {297--325},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {64},
number = {1},
year = {2014},
doi = {10.5802/aif.2848},
zbl = {06387275},
mrnumber = {3330550},
language = {en},
url = {http://www.numdam.org/articles/10.5802/aif.2848/}
}
TY - JOUR AU - Charpentier, Stéphane AU - Menet, Quentin AU - Mouze, Augustin TI - Closed universal subspaces of spaces of infinitely differentiable functions JO - Annales de l'Institut Fourier PY - 2014 SP - 297 EP - 325 VL - 64 IS - 1 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2848/ DO - 10.5802/aif.2848 LA - en ID - AIF_2014__64_1_297_0 ER -
%0 Journal Article %A Charpentier, Stéphane %A Menet, Quentin %A Mouze, Augustin %T Closed universal subspaces of spaces of infinitely differentiable functions %J Annales de l'Institut Fourier %D 2014 %P 297-325 %V 64 %N 1 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2848/ %R 10.5802/aif.2848 %G en %F AIF_2014__64_1_297_0
Charpentier, Stéphane; Menet, Quentin; Mouze, Augustin. Closed universal subspaces of spaces of infinitely differentiable functions. Annales de l'Institut Fourier, Tome 64 (2014) no. 1, pp. 297-325. doi : 10.5802/aif.2848. http://www.numdam.org/articles/10.5802/aif.2848/
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