Local dual spaces of Banach spaces of vector-valued functions

Abstract

We show that L∞ (μ, X* ) is a local dual of L 1 (μ,X ), and L 1 (μ, X*) is a local dual of L∞ (μ, X), where X is a Banach space. A local dual space of a Banach space Y is a subspace Z of Y* so that we have a local representation of Y* in Z satisfying the properties of the representation of X** in X provided by the principle of local reflexivity.