Afr. Mat. DOI 10.1007/s13370-014-0287-2 On paranorm I-convergent sequence spaces defined by a compact operator Vakeel A. Khan · Mohd Shafiq · Bernardo Lafuerza-Guillen Received: 26 November 2013 / Accepted: 21 August 2014 © African Mathematical Union and Springer-Verlag Berlin Heidelberg 2014 Abstract In this article we introduce and study paranorm I -convergent sequence spaces S I ( p), S I 0 ( p) and S I ∞ ( p) with the help of compact operator T on the real space R and a bounded sequence p = ( p k ) of positive real numbers. We study some topological and algebraic properties, prove the decomposition theorem and study some inclusion relations on these spaces. Keywords Compact operator · Ideal · Filter · I -convergent sequence · Solid and monotone space · Banach space · Paranorm Mathematics Subject Classification 41A10 · 41A25 · 41A36 · 40A30 1 Introduction and preliminaries Let N, R and C be the sets of all natural, real and complex numbers, respectively. We denote the space of all real or complex sequences by ω ={x = (x k ) : x k ∈ R or C}. Let ℓ ∞ , c and c 0 be denote the Banach spaces of bounded, convergent and null sequences of reals, respectively with norm V. A. Khan (B ) · M. Shafiq Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India e-mail: vakhanmaths@gmail.com M. Shafiq e-mail: shafiqmaths7@gmail.com B. Lafuerza-Guillen Department of Statistics and Applied Mathematics, University of Almeria, 04120 Almeria, Spain e-mail: blafuerz@ual.es 123