The Silverman–Toeplitz, Schur, and Steinhaus Theorems for Four-Dimensional Infinite Matrices

P. N. Natarajan ()
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P. N. Natarajan: Ramakrishna Mission Vivekananda College, Formerly of the Department of Mathematics

Chapter Chapter 5 in Classical Summability Theory, 2017, pp 83-100 from Springer

Abstract: Abstract In this chapter, aSteinhaus theorem new definitionSteinhaus theorem for 4-dimensional infinite matrices 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 theoremSilverman-Toeplitz theorem for 4-dimensional infinite matrices for four-dimensional infinite matrices is proved. We also proveSchur’s theorem for 4-dimensional infinite matrices Schur and Steinhaus theorems for four-dimensional infinite matrices.

Keywords: 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 (search for similar items in EconPapers)
Date: 2017
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-981-10-4205-8_5

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DOI: 10.1007/978-981-10-4205-8_5

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