On paranormed Ideal convergent sequence spaces defined by Jordan totient function

Vakeel A Khan,Umme Tuba

doi:10.1186/s13660-021-02634-7

Abstract

The study of sequence spaces and summability theory has been an important aspect in defining new notions of convergence for the sequences that do not converge in the usual sense. Paving the way into the applications of law of large numbers and theory of functions, it has proved to be an essential tool. In this paper we generalise the classical Maddox sequence spaces c_{0}(p), c(p), ell (p) and ell _{infty }(p) and define new ideal paranormed sequence spaces c^{I}_{0}(Upsilon ^{r}, p), c^{I}(Upsilon ^{r}, p), ell ^{I}_{ infty }(Upsilon ^{r}, p) and ell _{infty }(Upsilon ^{r}, p) defined with the aid of Jordan’s totient function and a bounded sequence of positive real numbers. We develop isomorphism between certain maps and also find their α-, β- and γ-duals. We examine algebraic and topological properties of these corresponding spaces. Further we study some standard inclusion relations and prove the decomposition theorem.

Highlights

We develop isomorphism between certain maps and find their α, βand γ -duals
Simons [1] generalised the classical bounded sequence spaces defined by Maddox [2] and Nakano [3] using a sequence of positive real numbers bounded above by 1
The most commonly used linear transformation on a sequence space is given by an infinite matrix

Summary

Introduction

∞(Υ r, p) defined with the aid of Jordan’s totient function and a bounded sequence of positive real numbers. The most commonly used linear transformation on a sequence space is given by an infinite matrix. Let M = (mst) be an infinite matrix of real numbers for s, t ∈ N and X and Y be two sequence spaces.

Results

Conclusion