General Summability Theory and Steinhaus Type Theorems

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

Chapter Chapter 1 in Classical Summability Theory, 2017, pp 1-26 from Springer

Abstract: Abstract In this chapter, we recall well-known definitions and concepts. We stateSilverman-Toeplitz theorem and prove Silverman–Toeplitz theorem and Schur’s theoremSchur’s theorem and then deduceSteinhaus theorem Steinhaus theorem. A sequenceSequence space $$\Lambda _r$$ space $$\Lambda _r$$ , $$r \ge 1$$ being a fixed integer, is introduced, and we make a detailed study of the space $$\Lambda _r$$ , especially from the point of view of sequences of zeros and ones. We prove a Steinhaus type result involving the space $$\Lambda _r$$ , which improves Steinhaus theorem. Some more Steinhaus type theorems are also proved.

Keywords: Infinite matrix; Banach space; Convergence preserving or conservative matrix; Regular matrix; Silverman–Toeplitz theorem; Schur’s theorem; Steinhaus theorem; Steinhaus type theorem; Sequence space $$\Lambda _r$$; Eventually periodic sequence; Non-periodic sequence; Sequence of zeros and ones; Closed linear span; Generalized semiperiodic sequence (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_1

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DOI: 10.1007/978-981-10-4205-8_1

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