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: Any two-dimensional Normed space is a generalized Day-James space | Alonso Journal of Inequalities and Applications 2011 2011 2 http content 2011 1 2 Journal of Inequalities and Applications a SpringerOpen Journal RESEARCH Open Access Any two-dimensional Normed space is a generalized Day-James space Javier Alonso Correspondence jalonso@ Department of Mathematics University of Extremadura 06006 Badajoz Spain Abstract It is proved that any two-dimensional normed space is isometrically isomorphic to a generalized Day-James space lv-lộ introduced by W. Nilsrakoo and S. Saejung. Keywords Normed space Day-James space Birkhoff orthogonality SpringerOpen0 1991 Mathematics Subject Classification 46B20 The Day-James space Ip-lq is defined for 1 p q - as the space R2 endowed with the norm x i i x pifX1X2 0 p q I x q if X1X2 0 where x x1 x2 . James 1 considered the space lp - lq with 1 p 1 q 1 as an example of a two-dimensional normed space where Birkhoff orthogonality is symmetric. Recall that if x and y are vectors in a normed space then x is said to be Birkhoff orthogonal to y x By if x ly x for every scalar l 2 . Birkhoff orthogonality coincides with usual orthogonality in inner product spaces. In arbitrary normed spaces Birkhoff orthogonality is in general not symmetric . in R2 with - P and it is symmetric in a normed space of three or more dimension if and only if the norm is induced by an inner product. This last significant property was obtained in gradual stages by Birkhoff 2 James 1 3 and Day 4 . The first reference related to the symmetry of Birkhoff orthogonality in two-dimensional spaces seems to be Radon 5 in 1916. He considered plane convex curves with conjugate diameters as in ellipses in order to solve certain variational problems. The procedure that James used to get two-dimensional normed spaces where Birkhoff orthogonality is symmetric was extended by Day 4 in the following way. Let X - X be a two-dimensional normed space and let u v e X be such that u X v X 1 u B v and
