ON THE QUASI-EQUIVALENCE
OF ORTHOGONAL BASES IN NON-ARCHIMEDEAN METRIZABLE LOCALLY CONVEX SPACES

WIESAW SLIWA

Faculty of Mathematics and Computer Science

A. Mickiewicz University

ul. Matejki 48/49, 60-769 Poznan

POLAND

sliwa@amu.edu.pl

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 A₁ (a) and $A_{\infty} (a) ,$ the most known and important examples of non-archimedean nuclear Fréchet spaces, have the quasi-equivalence property. It is easy to see that any two bases in c₀ are semi equivalent and any two bases in ${I\!K}^{I\!N}$ are equivalent. We show that the Fréchet spaces: $c_0 \times {I\!K}^{I\!N}$ and $c_0^{I\!N}$ have the quasi-equivalence property.