The Fixed Point Property in ð ‘ ð ŸŽ with an Equivalent Norm

We study the fixed point property (FPP) in the Banach space ð ‘ 0 with the equivalent norm ‖ â‹… ‖ ð · . The space ð ‘ 0 with this norm has the weak fixed point property. We prove that every infinite-dimensional subspace of ( ð ‘ 0 , ‖ â‹… ‖ ð · ) contains a complemented asymptotically isometric copy of ð ‘ 0 , and thus does not have the FPP, but there exist nonempty closed convex and bounded subsets of ( ð ‘ 0 , ‖ â‹… ‖ ð · ) which are not 𠜔 -compact and do not contain asymptotically isometric ð ‘ 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 ( ð ‘ 0 , ‖ â‹… ‖ ð · ) , and we give some of its properties. We also prove that the dual space of ( ð ‘ 0 , ‖ â‹… ‖ ð · ) over the reals is the Bynum space ð ‘™ 1 ∞ and that every infinite-dimensional subspace of ð ‘™ 1 ∞ does not have the fixed point property.