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  • Research Article
  • Open Access

A pythagorean approach in Banach spaces

Journal of Inequalities and Applications20062006:94982

  • Received: 30 December 2003
  • Accepted: 4 May 2004
  • Published:


Let be a Banach space and let be the unit sphere of . Parameters , , , and , where and are introduced and studied. The values of these parameters in the spaces and function spaces are estimated. Among the other results, we proved that a Banach space with , or is uniform nonsquare; and a Banach space with , or has uniform normal structure.


  • Banach Space
  • Function Space
  • Unit Sphere
  • Normal Structure
  • Uniform Normal Structure


Authors’ Affiliations

Department of Mathematics, Community College of Philadelphia, Philadelphia, PA 19130-3991, USA


  1. Brodskiĭ MS, Mil'man DP: On the center of a convex set. Doklady Akademii Nauk. SSSR (N.S.) 1948, 59: 837–840.MathSciNetGoogle Scholar
  2. Busemann H: The Geometry of Geodes. Academic Press, New York; 1955:x+422.Google Scholar
  3. Clarkson JA: Uniformly convex spaces. Transactions of the American Mathematical Society 1936,40(3):396–414. 10.1090/S0002-9947-1936-1501880-4MathSciNetView ArticleMATHGoogle Scholar
  4. Day MM: Normed Linear Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. Volume 21. 3rd edition. Springer, New York; 1973:viii+211.Google Scholar
  5. Gao J: Normal hexagon and more general Banach spaces with uniform normal structure. Journal of Mathematics. Shuxue Zazhi 2000,20(3):241–248.MATHMathSciNetGoogle Scholar
  6. Gao J: Normal structure, fixed points and related parameters in Banach spaces. Journal of Dynamical Systems and Geometric Theories 2002,1(1):1–18.MATHMathSciNetView ArticleGoogle Scholar
  7. Gao J, Lau K-S: On two classes of Banach spaces with uniform normal structure. Polska Akademia Nauk. Instytut Matematyczny. Studia Mathematica 1991,99(1):41–56.MATHMathSciNetGoogle Scholar
  8. García-Falset J: Stability and fixed points for nonexpansive mappings. Houston Journal of Mathematics 1994,20(3):495–506.MATHMathSciNetGoogle Scholar
  9. James RC: Uniformly non-square Banach spaces. Annals of Mathematics. Second Series (2) 1964, 80: 542–550. 10.2307/1970663MATHView ArticleMathSciNetGoogle Scholar
  10. Kirk WA: A fixed point theorem for mappings which do not increase distances. The American Mathematical Monthly 1965, 72: 1004–1006. 10.2307/2313345MATHMathSciNetView ArticleGoogle Scholar
  11. Mazcuñán-Navarro EM: On the modulus of-convexity of Ji Gao. Abstract and Applied Analysis 2003,2003(1):49–54. 10.1155/S1085337503204127MATHView ArticleMathSciNetGoogle Scholar
  12. Schäffer JJ: Geometry of Spheres in Normed Spaces, Lecture Notes in Pure and Applied Mathematics, no. 20. Marcel Dekker, New York; 1976:vi+228.Google Scholar
  13. Sims B: "Ultra"-Techniques in Banach Space Theory, Queen's Papers in Pure and Applied Mathematics. Volume 60. Queen's University, Ontario; 1982:iv+117.Google Scholar
  14. Sims B: A class of spaces with weak normal structure. Bulletin of the Australian Mathematical Society 1994,49(3):523–528. 10.1017/S0004972700016634MATHMathSciNetView ArticleGoogle Scholar


© Gao 2006

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