Some Results on Strongly Cesáro Ideal Convergent Sequence Spaces

Mohammad Faisal Khan,Ghulam Shabbir

doi:10.1155/2020/8897155

Abstract

Some algebraic properties of Cesáro ideal convergent sequence spaces, C I and C 0 I , are studied in this article and some inclusion relations on these spaces are established.

Highlights

Consider the space ω 􏼈x: xk ∈ R or C􏼉 of all real and complex sequences, where R and C are, respectively, the sets of all real and complex numbers.Suppose that l∞, c, and c0 are the linear spaces of bounded, convergent, and null sequences, respectively, normed by‖x‖∞ sup􏼌􏼌􏼌􏼌xk􏼌􏼌􏼌􏼌, where, k ∈ N, (1) kN being the set of all natural numbers.A sequence space x of complex numbers is said to be (C, 1) summable to L ∈ C if for ρk 1/k 􏽐ki 1 xi, limkρk L. e sequence (C, 1) is called Cesaro summable sequence of complex numbers over C
A sequence space is monotone if it contains the canonical preimages of its step spaces
A of canonical preimage all elements in λXK, of a i.e., step y is space in the λXK is a set of preimages canonical preimage of λXKif and only if y is the canonical preimage of some x ∈λXK

Summary

Introduction

Consider the space ω 􏼈x (xk): xk ∈ R or C􏼉 of all real and complex sequences, where R and C are, respectively, the sets of all real and complex numbers. Suppose that l∞, c, and c0 are the linear spaces of bounded, convergent, and null sequences, respectively, normed by. Let us denote by C1 the linear space of all (C, 1) summable sequences of complex numbers over C, i.e.,. Hardy and Littlewood [1] initiated the notion of strong Cesaro convergence for real numbers which is defined as follows. A sequence (xk) on a normed space (X, ‖ · ‖|) is said to be strongly Cesaro convergent to L if lim n⟶∞. Further interesting properties of Cesaro Ideal Convergent Sequences are established and a few inclusion relations are proved

Definitions of the Terms Used

Result

Main Results

Counter Example