The conjugate of a smooth Banach space

Abstract

A Banach space X is smooth if at every point of the unit sphere there is only one supporting hyperplane of the unit ball; and strictly convex, or rotund, if the unit sphere contains no line segment.Although there is a strong duality between these notions, Klee has produced a smooth space whose conjugate is not rotund. However there is no known example of a smooth space with conjugate not isomorphic to a rotund space.The main purpose of this note is to show that if X is a smooth space with a certain property, X* is isomorphic to a rotund space. This will follow from a mapping theorem which implies the existence of a set Γ and a continuous one-to-one linear map T of X* into co(Γ).