International Journal of Pure and Applied Mathematics ————————————————————————– Volume 68 No. 1 2011, 1-11 ON NON-STANDARD n-NORM ON SOME SEQUENCE SPACES Hemen Dutta 1 , B. Surender Reddy 2 § 1 Department of Mathematics Gauhati University Kokrajhar Campus, Assam, INDIA e-mail: hemen dutta08@rediffmail.com 2 Department of Mathematics Post Graduate College of Science - PGCS, Saifabad Osmania University Hyderabad, 500004, AP, INDIA e-mail: bsrmathou@yahoo.com Abstract: In this paper, we construct some difference sequence spaces which we call the spaces of Δ s (r) -convergent, Δ s (r) -null and Δ s (r) -bounded sequences with respect to n-norm on a real linear space X. We study these spaces by defining non-standard n-norm and (n − r)-norm for every r =1, 2 ...,n − 1. We show that under certain cases, convergence and completeness in the n-norm is equivalent to those in the (n −r)-norm. We also prove the fixed point theorem for these spaces, which are n-Banach spaces. AMS Subject Classification: 40A05, 46A45, 46B70 Key Words: n-normed spaces, completeness, fixed point theorem 1. Introduction Let w, ℓ ∞ , c and c 0 denote the spaces of all, bounded, convergent and null se- quences x =(x k ) with complex terms respectively, normed by ‖x‖ ∞ = sup k |x k |, where k ∈ N = {1, 2, 3,... } – the set of positive integers. Received: October 22, 2009 c  2011 Academic Publications § Correspondence author