In this chapter, we recall well-known definitions and concepts. We state and prove Silverman–Toeplitz theorem and Schur’s theorem and then deduce Steinhaus theorem. A sequence 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.
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
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