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1. CMB 2004 (vol 47 pp. 108)
On Universal Schauder Bases in Non-Archimedean FrÃ©chet Spaces 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 Categories:46S10, 46A35 |