Criterion for regular matrices in non-Archimedean fields. (English) Zbl 0751.40003

In this paper \(K\) denotes a complete, non-trivially valued, non- archimedean field and infinite matrices and sequences have entries in A. F. Monna [Nederl. Akad. Wet. Proc. Ser. A 66, 121–131 (1963; Zbl 0121.32703)] and J. B. Roberts [Proc. Am. Math. Soc. 8, 541–543 (1957; Zbl 0078.05003)] proved the criteria for convergence preservation and regularity of infinite matrices in \(K\) using non-archimedean functional analysis in the form of the analogue to the Banach-Steinhaus theorem. The purpose of the present paper is to prove these criteria for infinite matrices in \(K\) without recourse to non-archimedean functional analytic tools, using a technique of Schur, later fortified by V. Ganapathy Iyer in the case of the fields \(\mathbb{R}\) on \(\mathbb{C}\).
Reviewer: P.N.Natarajan


40C05 Matrix methods for summability
46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis