# General Summability Theory and Steinhaus Type Theorems

Chapter

## Abstract

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.

### 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

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