Journal of Inequalities and Applications
Volume 2006 (2006), Article ID 94982, 11 pages

A pythagorean approach in Banach spaces

Ji Gao

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

Received 30 December 2003; Accepted 4 May 2004

Copyright © 2006 Ji Gao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let X be a Banach space and let S(X)={xX,x=1} be the unit sphere of X. Parameters E(X)=sup{α(x),xS(X)}, e(X)=inf{α(x),xS(X)}, F(X)=sup{β(x),xS(X)}, and f(X)=inf{β(x),xS(X)}, where α(x)=sup{x+y2+xy2,yS(X)} and β(x)=inf{x+y2+xy2,yS(X)} are introduced and studied. The values of these parameters in the lp spaces and function spaces Lp[0,1] are estimated. Among the other results, we proved that a Banach space X with E(X)<8, or f(X)>2 is uniform nonsquare; and a Banach space X with E(X)<5, or f(X)>32/9 has uniform normal structure.