An extension of Phelps theorem to spaces of vector-valued functions

Abstract

In this paper, our main aim is to extend a classical theorem of Phelps on norm-attaining functionals from the space of scalar-valued continuous functions $C(\Omega)$ to its vector-valued counterpart $C(\Omega, X)$. One of our main results provides a complete characterization of norm-attaining functionals on $C(\Omega, X)$ under the assumption that $X^*$ has the Radon-Nikod\'ym property (RNP). For a general Banach space $X$, we further investigate norm attainment at points of weak$^*$-to-weak continuity for the identity map $Id : (C(\Omega, X)_1^*, w^*) \to (C(\Omega, X)_1^*, w)$.