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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