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On Universal Schauder Bases in Non-Archimedean Fréchet Spaces

  Published:2004-03-01
 Printed: Mar 2004
  • Wiesław Śliwa
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Abstract

It is known that any non-archimedean Fr\'echet space of countable type is isomorphic to a subspace of $c_0^{\mathbb{N}}$. In this paper we prove that there exists a non-archimedean Fr\'echet space $U$ with a basis $(u_n)$ such that any basis $(x_n)$ in a non-archimedean Fr\'echet space $X$ is equivalent to a subbasis $(u_{k_n})$ of $(u_n)$. Then any non-archimedean Fr\'echet space with a basis is isomorphic to a complemented subspace of $U$. In contrast to this, we show that a non-archimedean Fr\'echet space $X$ with a basis $(x_n)$ is isomorphic to a complemented subspace of $c_0^{\mathbb{N}}$ if and only if $X$ is isomorphic to one of the following spaces: $c_0$, $c_0 \times \mathbb{K}^{\mathbb{N}}$, $\mathbb{K}^{\mathbb{N}}$, $c_0^{\mathbb{N}}$. Finally, we prove that there is no nuclear non-archimedean Fr\'echet space $H$ with a basis $(h_n)$ such that any basis $(y_n)$ in a nuclear non-archimedean Fr\'echet space $Y$ is equivalent to a subbasis $(h_{k_n})$ of $(h_n)$.
Keywords: universal bases, complemented subspaces with bases universal bases, complemented subspaces with bases
MSC Classifications: 46S10, 46A35 show english descriptions Functional analysis over fields other than ${\bf R}$ or ${\bf C}$ or the quaternions; non-Archimedean functional analysis [See also 12J25, 32P05]
Summability and bases [See also 46B15]
46S10 - Functional analysis over fields other than ${\bf R}$ or ${\bf C}$ or the quaternions; non-Archimedean functional analysis [See also 12J25, 32P05]
46A35 - Summability and bases [See also 46B15]
 

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