The fixed point property in [c.sub.0] with an equivalent norm

• Published in: Abstract and applied analysis
• Main Authors: de Buen, Berta Gamboa, Nunez-Medina, Fernando
• Format: Journal Article
• Language: English
• Published: John Wiley & Sons, Inc 01.01.2011
• Subjects: Banach spaces; Fixed point theory; Mexico
• ISSN: 1085-3375
• DOI: 10.1155/2011/574614

We study the fixed point property (FPP) in the Banach space [c.sub.0] with the equivalent norm [[parallel]*[parallel].sub.D]. The space [c.sub.0] with this norm has the weak fixed point property. We prove that every infinite-dimensional subspace of ([c.sub.0], [[parallel]*[parallel].sub.D]) contains a complemented asymptotically isometric copy of [c.sub.0], and thus does not have the FPP, but there exist nonempty closed convex and bounded subsets of ([c.sub.0], [[parallel]*[parallel].sub.D]) which are not w-compact and do not contain asymptotically isometric [c.sub.0]--summing basis sequences. Then we define a family of sequences which are asymptotically isometric to different bases equivalent to the summing basis in the space ([c.sub.0], [[parallel]*[parallel].sub.D]), and we give some of its properties. We also prove that the dual space of ([c.sub.0], [[parallel]*[parallel].sub.D]) over the reals is the Bynum space [l.sub.1[infinity]] and that every infinite-dimensional subspace of [l.sub.1[infinity]] does not have the fixed point property.