Polar topologies on sequence spaces in non-Archimedean analysis (Q2915801)

In 2001, the first author and \textit{M. Babahmed} [Proyecciones 20, No. 2, 217--240 (2001)] studied the fundamental properties of separated dualities \(\langle X,Y\rangle\) of non-Archimedean topological vector spaces \(X,Y\) over a complete non-Archimedean and non-trivially valued field \(K\). In the present paper, the authors consider two vector-valued sequence spaces, \(E(X)\) and \(E(Y)\), over \(X\) and \(Y\) respectively, such that \(E(Y) \subset E(X)^{\beta}:= \{ (y_n)_n \in \omega(X) : \lim_n \langle x_n, y_n \rangle= 0 \text{ for all } (x_n)_n \in E(X) \}\). Then, by using the original duality \(\langle X,Y\rangle\), the authors define a duality pair \(\langle E(X), E(Y)\rangle\) and study the polar topologies on \(E(X)\) associated to this duality, especially those that are compatible with it. Finally, they give several characterizations for the subsets of \(E(X)\) that are compact, \(c\)-compact, complete and AK-complete with respect to these polar topologies.NEWLINENEWLINERecall that the existence of non-trivial convex \(c\)-compact sets in a non-Archimedean locally convex space implies that the base field \(K\) is spherically complete. For non-spherically complete \(K\), the adequate non-Archimedean substitutes of \(c\)-compact sets are the compactoid and complete ones. It would be interesting to characterize the compactoid (compactoid complete) subsets of \(E(X)\) for its polar topologies.