The Differences Between Birkhoff and Isosceles Orthogonalities in Radon Planes
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Abstract
The notion of orthogonality for vectors in inner product spaces is simple, interesting and fruitful. When moving to normed spaces, we have many possibilities to extend this notion. We consider Birkhoff orthogonality and isosceles orthogonality. Recently the constants which measure the difference between these orthogonalities have been investigated. The usual orthognality in inner product spaces and isosceles orthogonality in normed spaces are symmetric. However, Birkhoff orthogonality in normed spaces is not symmetric in general. A two-dimensional normed space in which Birkhoff orthogonality is symmetric is called a Radon plane. In this paper, we consider the difference between Birkhoff and isosceles orthogonalities in Radon planes.
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How to Cite
Mizuguchi, H. (2017). The Differences Between Birkhoff and Isosceles Orthogonalities in Radon Planes. Extracta Mathematicae, 32(2), 173-208. Retrieved from https://publicaciones.unex.es/index.php/EM/article/view/2605-5686.32.2.173
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Section
Banach Spaces and Operator Theory
References
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[8] R.C. James, Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc. 61 (1947), 265 – 292.
[9] R.C. James, Inner product in normed linear spaces, Bull. Amer. Math. Soc. 53 (1947), 559 – 566.
[10] D. Ji, S. Wu, Quantitative characterization of the difference between Birkhoff orthogonality and isosceles orthogonality, J. Math. Anal. Appl. 323 (1) (2006), 1 – 7.
[11] A. Jiménez-Melado, E. Llorens-Fuster, E.M. Mazcuñán-Navarro, The Dunkl-Williams constant, convexity, smoothness and normal structure, J. Math. Anal. Appl. 342 (1) (2008), 298 – 310.
[12] H. Martini, K.J. Swanepoel, The geometry of Minkowski spaces – a survey, II, Expo. Math. 22 (2) (2004), 93 – 144.
[13] H. Martini, K.J. Swanepoel, Antinorms and Radon curves, Aequationes Math. 72 (1-2) (2006), 110 – 138.
[14] H. Mizuguchi, The constants to measure the differences between Birkhoff and isosceles orthogonalities, Filomat 30 (2016), 2761 – 2770.
[15] H. Mizuguchi, K.-S. Saito, R. Tanaka, On the calculation of the Dunkl-Williams constant of normed linear spaces, Cent. Eur. J. Math. 11 (7) (2013), 1212 – 1227.
[16] K.-I. Mitani, K.-S. Saito, Dual of two dimensional Lorentz sequence spaces, Nonlinear Anal. 71 (11) (2009), 5238 – 5247.
[17] K.-I. Mitani, K.-S. Saito, T. Suzuki, Smoothness of absolute norms on Cn , J. Convex Anal. 10 (1) (2003), 89 – 107.
[18] K.-I. Mitani, S. Oshiro, K.-S. Saito, Smoothness of ψ-direct sums of Banach spaces, Math. Inequal. Appl. 8 (1) (2005), 147 – 157.
[19] P.L. Papini, S. Wu, Measurements of differences between orthogonality types, J. Math. Aanl. Appl. 397 (1) (2013), 285 – 291.
[20] B.D. Roberts, On the geometry of abstract vector spaces, Tôhoku Math. J. 39 (1934), 42 – 59.
[21] K.-S. Saito, M. Kato, Y. Takahashi, Von Neumann – Jordan constant of absolute normalized norms on C2 , J. Math. Anal. Appl. 244 (2) (2000), 515 – 532.
[22] T. Szostok, On a generalization of the sine function, Glas. Mat. Ser. III 38(58) (1) (2003), 29 – 44.
[2] D. Amir, “Characterization of Inner Product Spaces”, Operator Theory: Advances and Applications 20, Birkhauser Verlag, Basel, 1986.
[3] V. Balestro, H. Martini, R. Teixeira, Geometric properties of a sine function extendable to arbitrary normed planes, Monatsh. Math. 182 (4) (2017), 781 – 800.
[4] G. Birkhoff, Orthogonality in linear metric spaces, Duke Math. J. 1 (2) (1935), 169 – 172.
[5] F.F. Bonsall, J. Duncan, “Numerical Ranges II”, London Mathematical Society Lecture Note Series 10, Cambridge University Press, New York-London, 1973.
[6] M.M. Day, Some characterizations of inner-product spaces, Trans. Amer. Math. Soc. 62 (1947), 320 – 337.
[7] R.C. James, Orthogonality in normed linear spaces, Duke Math. J. 12 (1945), 291 – 302.
[8] R.C. James, Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc. 61 (1947), 265 – 292.
[9] R.C. James, Inner product in normed linear spaces, Bull. Amer. Math. Soc. 53 (1947), 559 – 566.
[10] D. Ji, S. Wu, Quantitative characterization of the difference between Birkhoff orthogonality and isosceles orthogonality, J. Math. Anal. Appl. 323 (1) (2006), 1 – 7.
[11] A. Jiménez-Melado, E. Llorens-Fuster, E.M. Mazcuñán-Navarro, The Dunkl-Williams constant, convexity, smoothness and normal structure, J. Math. Anal. Appl. 342 (1) (2008), 298 – 310.
[12] H. Martini, K.J. Swanepoel, The geometry of Minkowski spaces – a survey, II, Expo. Math. 22 (2) (2004), 93 – 144.
[13] H. Martini, K.J. Swanepoel, Antinorms and Radon curves, Aequationes Math. 72 (1-2) (2006), 110 – 138.
[14] H. Mizuguchi, The constants to measure the differences between Birkhoff and isosceles orthogonalities, Filomat 30 (2016), 2761 – 2770.
[15] H. Mizuguchi, K.-S. Saito, R. Tanaka, On the calculation of the Dunkl-Williams constant of normed linear spaces, Cent. Eur. J. Math. 11 (7) (2013), 1212 – 1227.
[16] K.-I. Mitani, K.-S. Saito, Dual of two dimensional Lorentz sequence spaces, Nonlinear Anal. 71 (11) (2009), 5238 – 5247.
[17] K.-I. Mitani, K.-S. Saito, T. Suzuki, Smoothness of absolute norms on Cn , J. Convex Anal. 10 (1) (2003), 89 – 107.
[18] K.-I. Mitani, S. Oshiro, K.-S. Saito, Smoothness of ψ-direct sums of Banach spaces, Math. Inequal. Appl. 8 (1) (2005), 147 – 157.
[19] P.L. Papini, S. Wu, Measurements of differences between orthogonality types, J. Math. Aanl. Appl. 397 (1) (2013), 285 – 291.
[20] B.D. Roberts, On the geometry of abstract vector spaces, Tôhoku Math. J. 39 (1934), 42 – 59.
[21] K.-S. Saito, M. Kato, Y. Takahashi, Von Neumann – Jordan constant of absolute normalized norms on C2 , J. Math. Anal. Appl. 244 (2) (2000), 515 – 532.
[22] T. Szostok, On a generalization of the sine function, Glas. Mat. Ser. III 38(58) (1) (2003), 29 – 44.