In this chapter, a new definition of convergence of a double sequence and a double series is introduced and its properties are studied. In the context of this new definition, the Silverman–Toeplitz theorem for four-dimensional infinite matrices is proved. We also prove Schur and Steinhaus theorems for four-dimensional infinite matrices.
Limit of a double sequence Convergent double series Absolutely convergent double series Pringsheim’s definition of limit of a double sequence four-dimensional infinite matrix Regular matrix Silverman–Toeplitz theorem ds-complete or double sequence complete Schur’s theorem Steinhaus theorem
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Natarajan, P.N., Srinivasan, V.: Silverman-Toeplitz theorem for double sequences and series and its application to Nörlund means in non-archimedean fields. Ann. Math. Blaise Pascal 9, 85–100 (2002)MathSciNetCrossRefMATHGoogle Scholar
Kojima, T.: On the theory of double sequences. Tôhoku Math. J. 21, 3–14 (1922)Google Scholar