On the summability of pairs of sequences in a nonarchimedean field
Let \(K\) be a complete non-trivially valued nonarchimedean field and let \(S\) denote the vector space of all sequences \(x= (x_n)\) with \(x_n\in K\) for all \(n\in\mathbb{N}\). A sequence \((A_k)\) of infinite matrices \(A_k= [a^{(k)}_{nj}]\) with entries from \(K\) defines a bilinear map \(A\) of \(D(A):= \{(x,y)\in S\times S\mid \sum_n \sum_j a^{(k)}_{nj}x_ny_j\) converges for every \(k\in\mathbb{N}\}\) into \(S\). The author obtains sufficient conditions on \((A_k)\) in order that (1) \(A:c\times c\to c\), and (2) \(A:\ell_\infty\times c\to c\), where \(c\) and \(\ell_\infty\) are, respectively, the space of all convergent and all bounded sequences.
scientific article; zbMATH DE number 1417952