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On Complemented Subspaces of Non-Archimedean Power Series Spaces

Published online by Cambridge University Press:  20 November 2018

Wiesław Śliwa  and
Agnieszka Ziemkowska
Wiesław Śliwa
Affiliation:
Faculty of Mathematics and Computer Science, A. Mickiewicz University, ul. Umultowska 87, 61-614 Poznań, Poland email: sliwa@amu.edu.plaziemk@amu.edu.pl
Agnieszka Ziemkowska
Affiliation:
Faculty of Mathematics and Computer Science, A. Mickiewicz University, ul. Umultowska 87, 61-614 Poznań, Poland email: sliwa@amu.edu.plaziemk@amu.edu.pl
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Abstract

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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.

Keywords

46S1047S1046A35non-archimedean Köthe spacerange of a continuous linear mapSchauder basis
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 63 , Issue 5 , 18 October 2011 , pp. 1188 - 1200
Copyright
Copyright © Canadian Mathematical Society 2011

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