The Size of the Unit Sphere

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 c0, which are not isometric, are nearly isometric. In a recent paper (2) Michael Cambern shows not only that c and c0 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.