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

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 $C(\mathrm{\Omega })$ to its vector-valued counterpart $C(\mathrm{\Omega },X)$. One of our main results provides a complete characterization of norm-attaining functionals on $C(\mathrm{\Omega },X)$ under the assumption that $X^{\ast }$ has the Radon-Nikodým property (RNP). For a general Banach space $X$, we further investigate norm attainment at points of weak${}^{\ast }$-to-weak continuity for the identity map $Id:(C(\mathrm{\Omega },X{)}_1^{\ast },w^{\ast })\to (C(\mathrm{\Omega },X{)}_1^{\ast },w)$.