Asplund spaces C_k(X) beyond Banach spaces

2025/10/02 by Fabian, Marian, Kcakol, Jerzy, Leiderman, Arkady
#54G12 #FOS: Mathematics #Functional Analysis (math.FA) #General Topology (math.GN) #Primary: 46E10 #Secondary: 54C35

paper · doi:10.48550/arxiv.2510.01873

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.

Citations

• Asplund spaces and the finest locally convex topology
• On the product of Weak Asplund locally convex spaces
• Tameness and Rosenthal type locally convex spaces

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• A Banach space characterization of (sequentially) Ascoli spaces
• Note on Kadets Klee property and Asplund spaces
• The Ascoli property for function spaces
• Ascoli's theorem for pseudocompact spaces
• The Bi-Compact-Open Topology on C(X)