On block basic sequences in non-Archimedean Köthe spaces

Abstract

We prove that any two Schauder bases $(x_n)$ and $(y_n)$ in non-normable Köthe spaces $E$ and $F$ (over a non-Archimedean field $\mathbb K$) have block basic sequences $(u_n)$ and $(v_n)$, respectively, that are equivalent. Moreover we show that any Schauder basis in a non-normable Köthe space has a block basic sequence that is equivalent to the coordinate Schauder basis in some generalized power series space of infinite type; the generalized power series spaces are the most known and important examples of nuclear Köthe spaces. It follows that any two non-normable Köthe spaces $E$ and $F$, have closed subspaces $E_0$ and $F_0$, respectively, that are isomorphic to the same generalized power series space of infinite type $D_g(a,\infty)$ for some $g\in \Phi_c$ and $a\in \Gamma$.