The Size of the Unit Sphere

Source abstract

Banach ( 1 , pp. 242-243) defines, for two Banach spaces X and Y , a number ( X, Y ) = inf (log (‖ L ‖ ‖ L -1 ‖)), where the infimum is taken over all isomorphisms L of X onto F. He says that the spaces X and Y are nearly isometric if ( X, Y ) = 0 and asks whether the concepts of near isometry and isometry are the same; in particular, whether the spaces c and c 0 , which are not isometric, are nearly isometric. In a recent paper ( 2 ) Michael Cambern shows not only that c and c 0 are not nearly isometric but obtains the elegant result that for the class of Banach spaces of continuous functions vanishing at infinity on a first countable locally compact Hausdorff space, the notions of isometry and near isometry coincide.