Some more Steinhaus type theorems over valued fields (Q1301238)

Let \(A\) be an infinite matrix with entries from a complete non-trivially valued non-Archimedean field. The author finds necessary and sufficient conditions for \(A\) to define a bounded operator between various sequence spaces: \(\ell\to c_0\), \(c\to c_0\), \(\ell_\infty \to c_0\), and also for the relation \[ \sum\limits_{n=0}^\infty (Ax)_n=\sum\limits_{k=0}^\infty x_k,\quad x=(x_k)\in \ell. \] The latter problem is studied also for \(K=\mathbb R\) or \(\mathbb C\).