- Research Article
- Open access
- Published:
A pythagorean approach in Banach spaces
Journal of Inequalities and Applications volume 2006, Article number: 94982 (2006)
Abstract
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.
References
BrodskiÄ MS, Mil'man DP: On the center of a convex set. Doklady Akademii Nauk. SSSR (N.S.) 1948, 59: 837–840.
Busemann H: The Geometry of Geodes. Academic Press, New York; 1955:x+422.
Clarkson JA: Uniformly convex spaces. Transactions of the American Mathematical Society 1936,40(3):396–414. 10.1090/S0002-9947-1936-1501880-4
Day MM: Normed Linear Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. Volume 21. 3rd edition. Springer, New York; 1973:viii+211.
Gao J: Normal hexagon and more general Banach spaces with uniform normal structure. Journal of Mathematics. Shuxue Zazhi 2000,20(3):241–248.
Gao J: Normal structure, fixed points and related parameters in Banach spaces. Journal of Dynamical Systems and Geometric Theories 2002,1(1):1–18.
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.
GarcÃa-Falset J: Stability and fixed points for nonexpansive mappings. Houston Journal of Mathematics 1994,20(3):495–506.
James RC: Uniformly non-square Banach spaces. Annals of Mathematics. Second Series (2) 1964, 80: 542–550. 10.2307/1970663
Kirk WA: A fixed point theorem for mappings which do not increase distances. The American Mathematical Monthly 1965, 72: 1004–1006. 10.2307/2313345
Mazcuñán-Navarro EM: On the modulus of-convexity of Ji Gao. Abstract and Applied Analysis 2003,2003(1):49–54. 10.1155/S1085337503204127
Schäffer JJ: Geometry of Spheres in Normed Spaces, Lecture Notes in Pure and Applied Mathematics, no. 20. Marcel Dekker, New York; 1976:vi+228.
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.
Sims B: A class of spaces with weak normal structure. Bulletin of the Australian Mathematical Society 1994,49(3):523–528. 10.1017/S0004972700016634
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Gao, J. A pythagorean approach in Banach spaces. J Inequal Appl 2006, 94982 (2006). https://doi.org/10.1155/JIA/2006/94982
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/JIA/2006/94982