On Complemented Subspaces of Non-Archimedean Power Series Spaces

Śliwa, Wiesław ; Ziemkowska, Agnieszka

Canadian journal of mathematics, Tome 63 (2011) no. 5, pp. 1188-1200

Résumé

The non-archimedean power series spaces, ${{A}_{1}}(a)\,\text{and}\,{{A}_{\infty }}(b)$ , are the best known and most important examples of non-archimedean nuclear Fréchet spaces. We prove that the range of every continuous linear map from ${{A}_{p}}(a)\,\text{to}\,{{A}_{q}}(b)$ has a Schauder basis if either $p\,=\,1$ or $p\,=\,\infty $ and the set ${{M}_{b,a}}$ of all bounded limit points of the double sequence ( ${{({{b}_{i}}/{{a}_{j}})}_{i,j\in \mathbb{N}}}$ is bounded. It follows that every complemented subspace of a power series space ${{A}_{p}}(a)$ has a Schauder basis if either $p\,=\,1$ or $p\,=\,\infty $ and the set ${{M}_{a,a}}$ is bounded.

Détail

DOI : 10.4153/CJM-2011-018-0

Mots-clés : 46S10, 47S10, 46A35, non-archimedean Köthe space, range of a continuous linear map, Schauder basis