Asplund spaces Cₖ(X) beyond Banach spaces

This paper addresses the Asplund property for the space of continuous functions Cₖ(X) equipped with the compact-open topology, when X is an arbitrary Tychonoff space. Motivated by inconsistent definitions in prior literature extending the Asplund property beyond Banach spaces, we provide a unified and self-contained treatment of core results in this context. A characterization of the Asplund property for Cₖ(X) is established, alongside a review of classical results, including the Namioka--Phelps theorem and its implications. All proofs are presented in a self-contained manner and rely on standard techniques.