On the existence of basic sequences in non-archimedean locally convex spaces

Abstract

We prove that there exist non-archimedean (n.a.) locally convex spaces without basic orthogonal sequences, and even without Schauder basic sequences. Among other things any n.a. Köthe space with the weak topology has no basic orthogonal sequence. On the other hand, we show that the strong dual of any infinite-dimensional n.a. polar Fréchet space and any n.a. LF-space have basic orthogonal sequences.