Schauder basis in a locally \(K\)-convex space and perfect sequence spaces (Q2891229)

Let \(E\) be a non-archimedean locally convex space. Suppose \(E\) has a Schauder basis \((e_i)_i\). The authors transfer topologies between \(E\) and the perfect sequence space associated to \(E\). As a product they solve the following problem: Determine the compatible topologies on \(E\) for which \((e_i)_i\) is an equicontinuous Schauder basis.NEWLINENEWLINEAnother interesting question, now related to weak Schauder bases (i.e., Schauder bases for the weak topology), is the following: Is every weak Schauder basis of \(E\) a Schauder basis of \(E\)? \textit{J. Kąkol} proved in [``Weak bases in \(p\)-adic spaces'', Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. (8) 5, No. 3, 667--676 (2002; Zbl 1072.46051)] that this question has an affirmative answer if and only if every weakly convergent sequence in \(E\) converges in \(E\). The authors provide in this paper an alternative proof of this affirmative answer.NEWLINENEWLINEThe authors also recall the result proved by T. Gildsdorf and J. Kąkol 1999, stating that every weak Schauder basis in a polarly barrelled polar space (e.g. in a polar Banach space), is an orthogonal basic sequence. But it was unknown if a Banach space with a weak Schauder base, for which every weak Schauder base is a basic sequence, is polar. A positive answer to this question is given in the present paper.NEWLINENEWLINEFor more information on Schauder bases in non-archimedean locally convex spaces, see Chapter 9 of [\textit{C. Perez-Garcia} and \textit{W. H. Schikhof}, Locally convex spaces over non-Archimedean valued fields. Cambridge: Cambridge University Press (2010; Zbl 1193.46001)].