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1998 | 127 | 1 | 21-46
Tytuł artykułu

A universal modulus for normed spaces

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We define a handy new modulus for normed spaces. More precisely, given any normed space X, we define in a canonical way a function ξ:[0,1)→ ℝ which depends only on the two-dimensional subspaces of X. We show that this function is strictly increasing and convex, and that its behaviour is intimately connected with the geometry of X. In particular, ξ tells us whether or not X is uniformly smooth, uniformly convex, uniformly non-square or an inner product space.
Słowa kluczowe
Twórcy
  • Departamento de Matemáticas, Universidad de Extremadura, 06071 Badajoz, Spain, cabero@ba.unex.es
autor
  • Bâtiment 101-Mathématiques, Université de Lyon 1, Boulevard du 11 Novembre 1918, 69622 Villeurbanne Cedex, France, yost@jonas.univ-lyon1.fr
Bibliografia
  • [1] J. Alonso and C. Benítez, Some characteristic and non-characteristic properties of inner product spaces, J. Approx. Theory 55 (1988), 318-323.
  • [2] J. Alonso and A. Ullán, Moduli in normed linear spaces and characterization of inner product spaces, Arch. Math. (Basel) 59 (1992), 487-495.
  • [3] D. Amir, Characterizations of Inner Product Spaces, Birkhäuser, Basel, 1986.
  • [4] C. Benítez and M. del Río, Characterization of inner product spaces through rectangle and square inequalities, Rev. Roumaine Math. Pures Appl. 29 (1984), 543-546.
  • [5] C. Benítez and Y. Sarantopoulos, Characterization of real inner product spaces by means of symmetric bilinear forms, J. Math. Anal. Appl. 180 (1993), 207-220.
  • [6] G. Birkhoff, Orthogonality in linear metric spaces, Duke Math. J. 1 (1935), 169-172.
  • [7] F. Cabello Sánchez and J. M. F. Castillo, Isometries of finite dimensional normed spaces, Extracta Math. 10 (1995), 146-151.
  • [8] M. M. Day, Polygons circumscribed about closed convex curves, Trans. Amer. Math. Soc. 62 (1947), 315-319.
  • [9] M. M. Day, Some characterizations of inner-product spaces, ibid., 320-337.
  • [10] M. del Río and C. Benítez, The rectangular constant for two-dimensional normed spaces, J. Approx. Theory 19 (1977), 15-21.
  • [11] J. Gao and K. S. Lau, On two classes of Banach spaces with uniform normal structure, Studia Math. 99 (1991), 41-56.
  • [12] K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Univ. Press, Cambridge, 1990.
  • [13] V. I. Gurariĭ, On differential properties of the modulus of convexity of Banach spaces, Mat. Issled. 2 (1967), 141-148 (in Russian).
  • [14] R. E. Harrell and L. A. Karlovitz, Girths and flat Banach spaces, Bull. Amer. Math. Soc. 76 (1970) 1288-1291.
  • [15] J.-B. Hiriart-Urruty, A short proof of the variational principle for approximate solutions of a minimization problem, Amer. Math. Monthly 90 (1983), 206-207.
  • [16] R. C. James, Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc. 61 (1947), 265-292.
  • [17] R. C. James, Uniformly non-square Banach spaces, Ann. of Math. 80 (1964), 540-550.
  • [18] J. Joly, Caractérisations d'espaces hilbertiens au moyen de la constante rectangle, J. Approx. Theory 2 (1969), 301-311.
  • [19] M. I. Kadets, Proof of the topological equivalence of all separable infinite-dimensional Banach spaces, Functional Anal. Appl. 1 (1967), 53-62 (= Funktsional. Anal. i Prilozhen. 1 (1967), 61-70).
  • [20] M. A. Khamsi, Étude de la propriété du point fixe dans les espaces de Banach et les espaces métriques, thesis, Univ. Paris VI, 1987.
  • [21] V. I. Liokumovich, Existence of B-spaces with a non-convex modulus of convexity, Izv. Vyssh. Uchebn. Zaved. Mat. 12 (1973), 43-49 (in Russian).
  • [22] V. D. Milman and G. Schechtman, Asymptotic Theory of Finite Dimensional Normed Spaces, Lecture Notes in Math. 1200, Springer, Berlin, 1986.
  • [23] S. Prus, A remark on a theorem of Turett, Bull. Polish Acad. Sci. Math. 36 (1988), 225-227.
  • [24] K. Przesławski and D. Yost, Lipschitz retracts, selectors and extensions, Michigan Math. J. 42 (1995), 555-571.
  • [25] S. Rolewicz, On drop property, Studia Math. 85 (1986), 27-35.
  • [26] J. J. Schäffer, Inner diameter, perimeter, and girth of spheres, Math. Ann. 173 (1967), 59-79 and 79-82.
  • [27] B. Sims, unpublished seminar notes, Kent State Univ., Kent, Ohio, 1986.
  • [28] C. Zanco and A. Zucchi, Moduli of rotundity and smoothness for convex bodies, Boll. Un. Mat. Ital. B (7) 7 (1993), 833-855.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-smv127i1p21bwm
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