On paranormed Ideal convergent sequence spaces defined by Jordan totient function

Abstract

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 ) $c_{0}(p)$ , c ( p ) $c(p)$ , ℓ ( p ) $\ell (p)$ and ℓ ∞ ( p ) $\ell _{\infty }(p)$ and define new ideal paranormed sequence spaces c 0 I ( ϒ r , p ) $c^{I}_{0}(\Upsilon ^{r}, p)$ , c I ( ϒ r , p ) $c^{I}(\Upsilon ^{r}, p)$ , ℓ ∞ I ( ϒ r , p ) $\ell ^{I}_{ \infty }(\Upsilon ^{r}, p)$ and ℓ ∞ ( ϒ r , p ) $\ell _{\infty }(\Upsilon ^{r}, p)$ 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.

Keywords

• Jordan totient operator
• Paranorm
• α, β, γ duals
• I-convergence
• Solidity
• Convergence free