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

TL;DR

This paper extends Phelps' theorem on norm-attaining functionals from scalar to vector-valued continuous functions, providing characterizations under the Radon-Nikodým property and exploring weak* to weak continuity.

Contribution

It generalizes a classical theorem to vector-valued functions and characterizes norm-attaining functionals for spaces with the Radon-Nikodým property.

Findings

Complete characterization of norm-attaining functionals when X* has RNP

Investigation of norm attainment at weak*-to-weak points for general Banach spaces

Extension of Phelps' theorem to vector-valued continuous 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 $Id : (C(\Omega, X)_1^*, w^*) \to (C(\Omega, X)_1^*, w)$.