|
|
Article: A Pythagorean approach in Banach spaces.
- Article from:
- Journal of Inequalities and Applications
- Article date:
- January 1, 2006
- Author:
CopyrightCOPYRIGHT 2006 Hindawi Publishing Corp. This material is published under license from the publisher through the Gale Group, Farmington Hills, Michigan. All inquiries regarding rights should be directed to the Gale Group. (Hide copyright information)
|
Let X be a Banach space and let S(X) = {x [member of] X, [parallel]x[parallel] = 1} be the unit sphere of X. Parameters E(X) = sup{[alpha](x), x [member of] S(X)}, e(X) = inf{[alpha](x), x [member of] S(X)}, F(X) = sup{[beta]p(x), x [member of]S(X) }, and f (X) = inf JP(x), x[member of]S(X) }, where a(x) = sup { Ik + y 11' + I I x - y 11" y[member of] S(X)}, and [beta](x) = inf{[[parallel]x + y[parallel].sup.2] + [[parallel]x - y[parallel].sup.2], y [member of] S(X)} are introduced and studied. The values of these parameters in the [l.sub.p spaces and function spaces [L.sub.p] [0, 1] are estimated. Among the other results, we proved that a Banach space X with E(X)
<8, or f (X)> 2 is ...
<5, or f (X)>
<1 such that for any subset K as above, there exists [x.sub.0] [member of] K such that sup{[parallel][x.sub.0] - y[parallel]>