Smoothness in some Banach Spaces of Operators and Vector Valued Functions

Abstract

[EN] A well known criterion of Smulyan states that the norm parallel to.parallel to of a real Banach space X is Gateaux differentiable at x is an element of X if and only if there is x* is an element of S-X* which is w*-exposed by x in B-X* and that the norm is Frechet differentiable at x if and only if there is x* is an element of S-X* which is w*-strongly exposed in B-X* by x. We show that in this criterion B-X* can be replaced by a convenient smaller set, and we apply this extended criterion to characterize the points of Gateaux and Frechet differentiability of the norm in epsilon products of Banach spaces, extending previous work of Heinrich. As a consequence we get some results of smoothness of the norm in some Banach spaces of continuous and harmonic vector valued functions.