On closed subspaces with Schauder bases in non-archimedean Frechet spaces

Abstract

The main purpose of this paper is to prove that a non-archimedean Frechet space of countable type is normable (respectively nuclear; reflexive; a Monte1 space) if and only if any its closed subspace with a Schauder basis is normable (respectively nuclear; reflexive; a Monte1 space). It is also shown that any Schauder basis in a non-normable non-archimedean Frechet space has a block basic sequence whose closed linear span is nuclear. It follows that any non-normable non-archimedean Frechet space contains an infinite-dimensional nuclear closed subspace with a Schauder basis. Moreover, it is proved that a non-archimedean Frechet space E with a Schauder basis contains an infinite-dimensional complemented nuclear closed subspace with a Schauder basis if and only if any Schauder basis in E has a subsequence whose closed linear span is nuclear.