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Keywords:
characterizations of inner product spaces; orthogonality; moduli of Banach spaces
characterizations of inner product spaces; orthogonality; moduli of Banach spaces
Summary:
Some characterizations of inner product spaces in terms of Birkhoff orthogonality are given. In this connection we define the rectangular modulus $\mu_{_X}$ of the normed space $X$. The values of the rectangular modulus at some noteworthy points are well-known constants of $X$. Characterizations (involving $\mu_{_X})$ of inner product spaces of dimension $\geq 2$, respectively $\geq 3$, are given and the behaviour of $\mu_{_X}$ is studied.
References:
[1] Alonso J.: Ortogonalidad en espacios normados. Ph.D. Thesis, Universidad de Extremadura, 1984. MR 0823479
[2] Alonso J., Benitez C.: Some characteristic and non-characteristic properties of inner product spaces. J. Approx. Theory 55 (1988), 318-325. MR 0968938 | Zbl 0675.41047
[3] Amir D.: Characterizations of Inner Product Spaces. Birkhauser Verlag, Basel-Boston-Stuttgart, 1986. MR 0897527 | Zbl 0617.46030
[4] Baronti M.: Su alcune parametri degli spazi normati. Boll. Un. Mat. Ital. 5 (18)-B (1981), 1065-1085.
[5] Baronti M., Papini L.: Projections, skewness and related constants in real normed spaces. Math. Pannon. 3 (1992), 31-47. MR 1170536 | Zbl 0763.46008
[6] Benitez C., Przeslawski K., Yost D.: A universal modulus for normed spaces. Studia Math. 127 1 (1998), 21-46. MR 1488142 | Zbl 0909.46008
[7] Day M.M.: Uniform convexity in factor and conjugate spaces Ann. of Math. (2). 45 (1944), 375-385. MR 0010779
[8] Desbiens J.: Constante rectangle et bias d'un espace de Banach. Bull. Austral. Math. Soc. 42 (1990), 465-482. MR 1083283
[9] Desbiens J.: Sur les constantes de Thele et de Schäffer. Ann. Sci. Math. Québec 16 (2) (1992), 129-141. MR 1199184 | Zbl 0788.46018
[10] Franchetti C.: On the radial projection in Banach spaces. in Approximation Theory III (ed. by E.W. Cheney), pp.425-428, Academic Press, New York, 1980. MR 0602747 | Zbl 0483.46012
[11] James R.C.: Orthogonality and linear functionals in normed linear spaces. Trans. Amer. Math. Soc. 61 (1947), 265-292. MR 0021241 | Zbl 0037.08001
[12] Joly J.L.: Caracterizations d'espaces hilbertiens au moyen de la constante rectangle. J. Approx. Theory 2 (1969), 301-311. MR 0270126
[13] Lindenstrauss J.: On the modulus of smoothness and divergent series in Banach spaces. Michigan Math. J. 10 (1963), 241-252. MR 0169061 | Zbl 0115.10001
[14] Nordlander G.: The modulus of convexity in normed linear spaces. Ark. Mat. 4 (1960), 15-17. MR 0140915 | Zbl 0092.11402
[15] del Rio M., Benitez C.: The rectangular constant for two-dimensional spaces. J. Approx. Theory 19 (1977), 15-21. MR 0448039 | Zbl 0343.46018
[16] Şerb I.: On the behaviour of the tangential modulus of a Banach space I. Revue d'Analyse Numérique et de Théorie de l'Approximation 24 (1995), 241-248. MR 1608428
[17] Şerb I.: On the behaviour of the tangential modulus of a Banach space II. Mathematica (Cluj) 38 (61) (1996), 199-207. MR 1606805
[18] Şerb I.: A Day-Nordlander theorem for the tangential modulus of a normed space. J. Math. Anal. Appl. 209 (1997), 381-391. MR 1474615
[19] Smith M.A.: On the norms of metric projections. J. Approx. Theory 31 (1981), 224-229. MR 0624010 | Zbl 0477.46022
[20] Thele R.L.: Some results on the radial projection in Banach spaces. Proc. Amer. Math. Soc. 42 (1974), 2 483-486. MR 0328550 | Zbl 0276.46015
[21] Ullán de Celis A.: Modulos de convexidad y lisura en espacios normados. Ph.D. Thesis, Universidad de Extremadura, 1991. MR 1174971
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