Closed universal subspaces of spaces of infinitely differentiable functions

Charpentier, Stéphane; Menet, Quentin; Mouze, Augustin

Annales de l'Institut Fourier, Tome 64 (2014) no. 1, pp. 297-325.

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 ${𝕂}^{\mathbb{N}}$.

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 ${𝕂}^{\mathbb{N}}.$

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