On the quasi-equivalence of orthogonal bases in non-archimedean metrizable locally convex spaces

Abstract

We prove that any non-Archimedean metrizable locally convex space $E$ with a regular orthogonal basis has the quasi-equivalence property, i.e. any two orthogonal bases in $E$ are quasi-equivalent. In particular, the power series spaces $A_1(a)$ and $A_\infty(a)$, the most known and important examples of non-Archimedean nuclear Fréchet spaces, have the quasi-equivalence property. We also show that the Fréchet spaces: ${\Bbb K}^{\Bbb N},c_0\times{\Bbb K}^{\Bbb N},c^{\Bbb N}_0$ have the quasi-equivalence property.