International Journal of Mathematical Archive-7(4), 2016, 161-165 Available online through www.ijma.info ISSN 2229 – 5046 International Journal of Mathematical Archive- 7(4), April – 2016 161 ON EQUIVALENCE OF -NORMS AND THEIR DERIVED NORMS ON THE SPACE OF -SUMMABLE SEQUENCES PRADEEP KUMAR SINGH*, J. K. SRIVASTAVA Department of Mathematics and Statistics, D D U Gorakhpur University, Gorakhpur – India, 273009. (Received On: 07-04-16; Revised & Accepted On: 30-04-16) ABSTRACT In this paper, we shall investigate equivalence relation between - and   defined on   ;1 ≤ < ∞. Moreover, we shall show that non-equivalent - can also give equivalent  . Keywords: -summable sequences, parallel rearranged sequences,  , - ,  , equivalence. AMS Subject Classification: 40A05, 46A45, 46B20, 46B45, 46B99. 1. INTRODUCTION The concept of 2-  was initially investigated by Gähler [2] in the mid of 1960’s. Since then this theory has been developed in many directions including its generalization to - , for instance see Malćeski [7], Gunawan [1, 3, 4, 5, 6] and many others. Definition1.1: Let  be a vector space over (= ℝ or ℂ ) of dimension  ≥  (≥2). A non-negative real valued function ‖.,.,…,. ‖ defined on   satisfying the four conditions: (N 1 ) ‖  ,   ,…,   ‖ = 0 if and only if   ,   ,…,   are linearly dependent; (N 2 ) ‖  ,   ,…,   ‖ is invariant under any permutation of   ,   ,…,   ; (N 3 ) ‖  ,   ,…,   ‖ = ||‖  ,   ,…,   ‖; (N 4 ) ‖  + ,   ,…,   ‖ ≤ ‖  ,   ,…,   ‖ + ‖,   ,…,   ‖; for all   ,   ,…,   ,  ∈  and for all ∈, is called an -  on , and the pair ( , ‖.,.,…,. ‖) is called an -  . Definition 1.2: Let ( , ‖.,.,…,. ‖) be an -  . A sequence �  �  =0 ∞ in  is said to be a Cauchy sequence if �  − ′ ,   ,   ,...,  − � → 0 as ,  ′ → ∞ for all   ,   ,…,  − ∈. �  �  =0 ∞ is said to be converges at ∈ ( is called limit point of the sequence) if �  − ,   ,   ,...,  − �→ 0 as → ∞ for all   ,   ,…,  − ∈ . An -   ( , ‖.,.,…,. ‖) is called an -  (  - ) if every Cauchy sequence in  converges to an element of . A vector space can be equipped with several - , see [1]. In such a case, our aim is to investigate an equivalence relation between them. Definition 1.3: Two  ‖ . ‖ 1 and ‖ . ‖ 2 defined on a linear space  are said to be equivalent if there exists  >0 and  >0 such that: ‖‖ 1 ≤ ‖‖ 2 ≤ ‖‖ 1 for all  ∈ . Similar to the above definition, two -  ‖.,.,…,. ‖ 1 and ‖.,.,…,. ‖ 2 defined on a linear space  are said to be equivalent if there exists  >0 and  >0 such that: ‖  ,   ,…,   ‖ 1 ≤ ‖  ,   ,…,   ‖ 2 ≤ ‖  ,   ,…,   ‖ 1 for all   ,   ,…,   ∈ . -Norms which are not equivalent are termed as non-equivalent (or un-equivalent). Now we state the results of [6, 8] as following Lemma: Corresponding Author: Pradeep Kumar Singh*. E-mail: pradeep3789@gmail.com.