General Summability Theory and Steinhaus Type Theorems

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.