ISSN 1995-0802, Lobachevskii Journal of Mathematics, 2012, Vol. 33, No. 2, pp. 183–190. c  Pleiades Publishing, Ltd., 2012. Some Double Sequence Spaces Defined by a Sequence of Orlicz Functions Over n-Normed Spaces Kuldip Raj * , Ajay K. Sharma ** , Sunil K. Sharma *** , and Sulinder Singh (Submitted by D.H. Mushtari) School of Mathematics Shri Mata Vaishno Devi University, Katra, 182320 J&K, India Received November 17, 2011 Abstract—In the present paper we introduced double sequence spaces defined by a sequence of Orlicz functions M =(M k,l ) over n-normed spaces and examine some properties of the resulting sequence spaces. DOI: 10.1134/S1995080212020060 Keywords and phrases: Orlicz function, Musielak-Orlicz function, n-normed space, paranorm space, double sequence space, solid, monotone. 1. INTRODUCTION AND PRELIMINARIES The concept of 2-normed spaces was initially developed by G ¨ ahler [4] in the mid of 1960’s, while that of n-normed spaces one can see in Misiak [11]. Since then, many others have studied this concept and obtained various results, see Gunawan ([5], [6]) and Gunawan and Mashadi [7]. Let n ∈ N and X be a linear space over the field K, where K is field of real or complex numbers of dimension d, where d ≥ n ≥ 2. A real valued function ||·, ··· , ·|| on X n satisfying the following four conditions: (1) ||x 1 ,x 2 , ··· ,x n || =0 if and only if x 1 ,x 2 , ··· ,x n are linearly dependent in X; (2) ||x 1 ,x 2 , ··· ,x n || is invariant under permutation; (3) ||αx 1 ,x 2 , ··· ,x n || = |α|||x 1 ,x 2 , ··· ,x n || for any α ∈ K, and (4) ||x + x ′ ,x 2 , ··· ,x n || ≤ ||x, x 2 , ··· ,x n || + ||x ′ ,x 2 , ··· ,x n || is called a n-norm on X, and the pair (X, ||·, ··· , ·||) is called a n-normed space over the field K. For example, we may take X = R n being equipped with the Euclidean n-norm ||x 1 ,x 2 , ··· ,x n || E = the volume of the n-dimensional parallelopiped spanned by the vectors x 1 ,x 2 , ··· ,x n which may be given explicitly by the formula ||x 1 ,x 2 , ··· ,x n || E = | det(x ij )|, where x i =(x i1 ,x i2 , ··· ,x in ) ∈ R n for each i =1, 2, ··· ,n. Let (X, ||·, ··· , ·||) be a n-normed space of dimension d ≥ n ≥ 2 and {a 1 ,a 2 , ··· ,a n } be linearly independent set in X. Then the following function ||·, ··· , ·|| ∞ on X n−1 defined by ||x 1 ,x 2 , ··· ,x n || ∞ = max{||x 1 ,x 2 , ··· ,x n−1 ,a i || : i =1, 2, ··· ,n} defines an (n − 1)-norm on X with respect to {a 1 ,a 2 , ··· ,a n }. A sequence (x k ) in a n-normed space (X, ||·, ··· , ·||) is said to converge to some L ∈ X if lim k→∞ ||x k − L, z 1 , ··· ,z n || =0 for every z 1 , ··· ,z n ∈ X. * E-mail: kuldeepraj68@rediffmail.com ** E-mail: aksju_76@yahoo.com *** E-mail: sunilksharma42@yahoo.co.in 183