tailieunhanh - Báo cáo hóa học: " Research Article Some Properties of Pythagorean Modulus"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Some Properties of Pythagorean Modulus | Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2007 Article ID71012 6 pages doi 2007 71012 Research Article Some Properties of Pythagorean Modulus Fenghui Wang Received 7 July 2007 Revised 22 November 2007 Accepted 3 December 2007 Recommended by Andras Ronto We consider two Pythagorean modulus introduced by Gao 2005 2006 recently. The exact values concerning these modulus for some classical Banach spaces are determined. Some applications in geometry of Banach spaces are also obtained. Copyright 2007 Fenghui Wang. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction Recently Gao introduced some moduli from Pythagorean theorem. In terms of these moduli he got some sufficient conditions for a Banach space X to have uniform normal structure which plays an import role in fixed-point theory. In this paper we mainly discuss the moduli Ee X and fe X . Let X be a Banach space. By SX and BX we will denote the unit sphere and unit ball of X respectively. For every nonnegative number e the Pythagorean moduli are given by 1 2 Ee X sup x ey 2 x- ey 2 x y e Sx fe X inf x ey 2 x- ey 2 x y e Sx . For simplicity we will write E e and f e for Ee X and f X provided no confusion occurs. It is clear that 2 f e 2 1 e2 E e 2 1 e 2. It is also worth noting that the first moduli Ee X has been proved to be very useful in the study of the well-known von Neumann-Jordan constant see . 3 4 . Following Gao we study the further properties concerning the Pythagorean moduli. We find that these moduli are connected with some geometric properties. They enable us to distinguish several important classes of spaces such as uniformly convex uniformly smooth or uniformly nonsquare. 2 Journal of Inequalities and Applications 2. Pythagorean modulus We can replace SX by BX in the definition of