Asplund spaces $C_k(X)$ beyond Banach spaces

TL;DR

This work characterizes when the locally convex space $C_k(X)$, endowed with the compact-open topology, is Asplund for arbitrary Tychonoff $X$, unifying prior definitions and extending the Namioka–Phelps framework. It proves a comprehensive set of equivalences: $C_k(X)$ is Asplund; every compact subset of $X$ is scattered; $C_k(X)$ contains no copy of $oldsymbol{ extell}_1$; every separable Banach subspace of $C_k(X)$ has a separable dual; $C_k(X)$ has the Namioka–Phelps property; and $C_k(X)$ is tame. The results hinge on separable reductions, bound-covering embeddings into $C(K)$ spaces, and the role of $oldsymbol{ riangle_1}$-spaces, establishing that for $oldsymbol{ extomega}$-bounded $X$ the Asplund property of $C_k(X)$ is equivalent to $X$ being scattered (and to every compact subset being scattered, as well as to $X$ being a $oldsymbol{ riangle_1}$-space). The final remarks include examples showing that Asplundness behaves differently in locally convex spaces than in Banach spaces, emphasizing limitations of transferring Banach-space intuition to the $C_k(X)$ setting.

Abstract

This paper addresses the Asplund property for the space of continuous functions $C_k(X)$ 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 $C_k(X)$ 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.

Theorems & Definitions (39)

• Proposition 1.1
• Lemma 1.2
• Theorem 1.3
• Theorem 1.4
• Proposition 1.5
• Proposition 1.6
• proof
• Remark 1.7
• Definition 1.8
• Definition 1.9