International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Volume 2, Issue 7, July 2014, PP 668-674 ISSN 2347-307X (Print) & ISSN 2347-3142 (Online) www.arcjournals.org ©ARC Page | 668 Strictly Convexity in Quasi 2-pre-Hilbert Spaces Risto Malčeski Faculty for informatics FON University Skopje, Macedonia risto.malceski@gmail.com Katerina Anevska Faculty for Informatics FON University Skopje, Macedonia anevskak@gmail.com Abstract: Using left-hand and right-hand Gateaux derivative of a 2-norm in [1] is given functional (,)() gxz y , which is generalization of 2-inner product and is used for defining a quasi 2-pre-Hilbert space. Further, in [1] is proved that each quasi 2-pre-Hilbert space is smooth. The strictly convexity in quasi 2-pre-Hilbert space is not object of interest in [1]. So, in this paper exactly that will be the focus of our interest. Keywords: 2-normed space, smooth space, quasi 2-pre-Hilbert space, strictly convex space 2010 Mathematics Subject Classification. 46B20, 46C05 1. INTRODUCTION Let ( ,|| , ||) L be a real 2-normed space. Then on L L L exist the functional || , || || , || 0 (,)() lim x ty z xz t t N xz y , || , || || , || 0 (,)() lim x ty z xz t t N xz y , (1) which are called as left-hand and right-hand Gateaux derivative of the 2-norm || , || at (,) xz in the direction y , respectively. Therefore, exists the functional || , || 2 (,)() ( (,)() ( , )( )) xz gxz y N xz y N xz y . (2) The functional (,)() gxz y is generalization of 2-inner product, and in 2-pre-Hilbert space corresponds to 2-inner product, Theorem 2, [1]. The Theorem 1, [1] proves that in each 2-normed space the following statements are true: 2 (,)() || , || gxz x xz , for every , xz L , (3) | ( , )( )| || , || || , || gxz y xz yz , for every ,, xyz L , (4) 2 ( , )( ) || , || (,)() gxz x y xz gxz y , for every ,, xyz L , (5) ( , )( ) (,)() g xz y gxz y , for every ,, ; , xyz L R , (6) || , || || , || || , || || , || || , || (,)() || , || x yz xz x ty z xz t xz gxz y xz , 0, 0, , , t xyz L . (7) In a 2-pre-Hilbert space hold the parallelepiped equality 2 2 2 2 || , || || , || 2(|| , || || , || ) x yz x yz xz yz , ,, xyz L (8) and the following one, which is equivalent to the parallelepiped equality 4 4 2 2 || , || || , || 8(|| , || || , ||)(, |) x yz x yz xz yz xyz , (9) (Lemma 2, [1]), and thus in 2-normed space the equality