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 Ck(X) endowed with the compact-open topology, where X is an arbitrary Tychonoff space. We completely characterize the Asplund property for Ck(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.
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BibTeX
@misc{fabian2026survey,
title = {{A survey on the Asplund property for spaces $C(X)$ of continuous functions}},
author = {Marian Fabian and Jerzy Kakol and Arkady Leiderman},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23167},
eprint = {2609.23167},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23167},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A survey on the Asplund property for spaces C(X) of continuous functions
AU - Marian Fabian
AU - Jerzy Kakol
AU - Arkady Leiderman
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.23167
UR - https://arxiv.org/abs/2609.23167
AB - 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.
ER -