On Universal Schauder Bases in Non-Archimedean Fréchet Spaces

Śliwa, Wiesław

Canadian mathematical bulletin, Tome 47 (2004) no. 1, pp. 108-118

Résumé

It is known that any non-archimedean Fréchet 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 Fréchet space $U$ with a basis $({{u}_{n}})$ such that any basis $({{x}_{n}})$ in a non-archimedean Fréchet space $X$ is equivalent to a subbasis $({{u}_{kn}})$ of $({{u}_{n}})$ . Then any non-archimedean Fréchet space with a basis is isomorphic to a complemented subspace of $U$ . In contrast to this, we show that a non-archimedean Fréchet 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 Fréchet space $H$ with a basis $({{h}_{n}})$ such that any basis $({{y}_{n}})$ in a nuclear non-archimedean Fréchet space $Y$ is equivalent to a subbasis $({{h}_{kn}})$ of $({{h}_{n}})$ .

Détail

DOI : 10.4153/CMB-2004-012-2

Mots-clés : 46S10, 46A35, universal bases, complemented subspaces with bases