The Fixed Point Property in c0 with an Equivalent Norm
We study the fixed point property (FPP) in the Banach space c0 with the equivalent norm ‖⋅‖D. The space c0 with this norm has the weak fixed point property. We prove that every infinite-dimensional subspace of (c0,‖⋅‖D) contains a complemented asymptotically isometric copy of c0, and thus does not h...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2011-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2011/574614 |
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| Summary: | We study the fixed point property (FPP) in the Banach space c0 with the equivalent norm ‖⋅‖D. The space c0 with this norm has the weak fixed point property. We prove that every infinite-dimensional subspace of (c0,‖⋅‖D) contains a complemented asymptotically isometric copy of c0, and thus does not have the FPP, but there exist nonempty closed convex and bounded subsets of (c0,‖⋅‖D) which are not ω-compact and do not contain asymptotically isometric c0—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 (c0,‖⋅‖D), and we give some of its properties. We also prove that the dual space of (c0,‖⋅‖D) over the reals is the Bynum space l1∞ and that every infinite-dimensional subspace of l1∞ does not have the fixed point property. |
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| ISSN: | 1085-3375 1687-0409 |