A survey on the Asplund property for spaces $C(X)$ of continuous functions

Abstract

The Asplund property plays an important role in Banach space theory due to its connections with differentiability properties of continuous convex functions, optimization problems, and the weak topology of Banach spaces. Motivated by the variety of nonequivalent definitions proposed in the literature for locally convex spaces, in this survey we provide a unified framework for studying the Asplund property beyond the Banach space setting. The main object of our study is the locally convex space of continuous functions $C_k(X)$ endowed with the compact-open topology, where $X$ is an arbitrary Tychonoff space. We completely characterize the Asplund property for $C_k(X)$ in terms of topological properties of the underlying space $X$. As an essential step, we revisit the proof of several classical results, including the Namioka--Phelps theorem. Our approach is independent of differentiability techniques and relies solely on topological methods. The exposition is self-contained, and all major results are provided with complete proofs, making the paper accessible to both specialists and newcomers.