\(k\)-spaces and duals of non-Archimedean metrizable locally convex spaces
A Hausdorff topological space \(X\) is called a \(k\)-space if a subset \(A\) of \(X\) is open (closed) if and only if \( A\cap C\) is open (closed) in \(C\) for every compact subset \(C\) of \(X\). If \(X\) is zero-dimensional, i.e., the topology admits a base formed of clopen (closed and open) sets, then it is called a \(k_0\)-space if and only if the above property holds for any clopen subset \(A\) of \(X\). The paper is concerned with the \(k\)-space properties of non-Archimedean locally convex spaces. Let \(\mathbb{K}\) be a complete non-Archimedean valued field and \(E\) a non-Archimedean locally convex space over \(\mathbb{K}\), with dual \(E'\) (see [\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)]). A subset \(A\) of \(E\) is called a compactoid if, for every neighborhood \(V\) of \(0\in E\), there exists a finite set \(Z\subset E\) such that \(A\subset V+\)aco\((Z)\), where aco\((Z)\) denotes the absolutely convex hull of \(Z\). One denotes by \(\sigma(E',E)\) (\(c(E',E)\)) the locally convex topology on \(E'\) of uniform convergence on all finite (compactoid) subsets of \(E\) and let \(E'_c\) be the space \(E'\) equipped with the topology \(c(E',E)\). If \(E\) is a metrizable and \(\mathbb{K}\) is locally compact, then \(E'_c\) is a \(k\)-space, hence a \(k_0\)-space. This result is no longer true if \(\mathbb{K}\) is not locally compact -- in this case, the space \(\left(\mathbb{K}^{\mathbb{N}}\right)'_c\) is not a \(k\)-space. In fact, the local compactness of \(\mathbb{K}\) is equivalent to the fact that, for every metrizable locally convex space \(E\) over \(\mathbb{K}\), the space \(E'_c\) is a \(k\)-space. The author also proves a non-Archimedean analog of a result of Dieudonné -- if \(E\) is metrizable and of countable type, then \(E'_c\) is a \(k\)-space if and only if \(c(E',E)\) is the finest topology on \(E'\) agreeing with \(\sigma(E',E)\) on equicontinuous subsets of \(E'\).
non-Archimedean Banach-Dieudonné theorem
scientific article; zbMATH DE number 6947026