International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Volume 2, Issue 6, June 2014, PP 603-610 ISSN 2347-307X (Print) & ISSN 2347-3142 (Online) www.arcjournals.org ©ARC Page | 603 About the Strictly Convex and Uniformly Convex Normed and 2-Normed Spaces Risto Malčeski Faculty for informatics FON University, Skopje, Macedonia risto.malceski@gmail.com Ljupcho Nastovski Faculty of Natural science Sts.”Cyril and Methodius”, Skopje, Macedonia ljupcona@pmf.ukim.mk Biljana Načevska Faculty of Electrical Engineering and Information Sts.”Cyril and Methodius”, Skopje, Macedonia biljanan@feit.ukim.edu.mk Admir Huseini Independent researcher, RISAT Skopje, Macedonia huseini@risat.org Abstract: In [1] A. Khan introduces the notion of uniformly convex 2-normed space and prove some properties of the uniformly convex 2-normed spaces. In this work, further properties of the uniformly convex 2-normed spaces are given and the question of the convexity of a normed space in which the norm is induced by 2-norm is analysed. Keywords: 2-normed space, 2-pre-Hibert space, convergent sequence, strictly convex space, uniformly convex space 2010 Mathematics Subject Classification. 46B20, 46C05 1. INTRODUCTION The concept of a uniformly convex 2-normed space is introduced by A. Khan. For our further investigation, we will introduce the definition of the uniformly convex 2-normed space in its equivalent form, as follows. Definition 1 ([1]). A 2-normed space ( ,|| , ||) L is uniformly convex if for every 0 there exists () 0 such that || , || || , || 1 xz yz , || , || x yz and (, ) z Vxy implies || , || 2(1 ( )) x yz , where (, ) Vxy is the subspace generated by the vectors x and y . Example 1 ([1]). A 2-pre-Hibert space is a 2-normed space in which the norm is introduced by 2 || , || (, | ) xy xxy and the parallelepiped law is satisfied 2 2 2 2 || , || || , || 2(|| , || || , || ) x yz x yz xz yz . (1) If 0 is given and || , || || , || 1 xz yz , || , || x yz and (, ) z Vxy , then from the equality (1) it follows that for 2 2 () 1 1 () 0 the following 2 1/2 2 1/2 || , || (4 || , || ) (4 ) 2(1 ( )) x yz x yz holds. It means, that ( ,(, | )) L is uniformly convex space. Let z be a fixed nonzero element in L , () Vz be the subspace of L generated by z and let z L be the quotient space / () LVz . For x L by z x we denote the class of equivalence of x over () Vz . Clearly, z L is a linear space with the operations of adding the two vectors and multiplying a