Examples of non-archimedean nuclear Fréchet spaces without a Schauder basis (Q5951551)

This paper presents a solution of a long-standing problem in \(p\)-adic functional analysis.NEWLINENEWLINENEWLINELet \(k\) be a non-archimedean, non-trivially valued field which is complete with respect to the metric induced by the valuation \(|\cdot |:K\to [0,\infty)\); let \(E\) be an infinite-dimensional Fréchet space of countable type, i.e., there is a countable set whose linear span is dense over \(K\). A sequence \(e_1,e_2,\dots,\) is called a Schauder basis of \(E\) if every \(x\in E\) has a unique expansion \(x=\sum^\infty_{n=1}\lambda_ne_n\), where \(\lambda_n\in K\), \(\lambda_ne_n\to 0\).NEWLINENEWLINENEWLINEIn the present paper, it is shown that not every \(E\) has a Schauder basis. Much more strongly, an infinite family of pairwise non-isomorphic nuclear \(E\)'s without Schauder basis is constructed. Also, an example that, in addition, has no bounded approximation property is presented. NEWLINENEWLINENEWLINEThe constructions are inspired by classical techniques used by Bessaga, Mitiagin, and Vogt.