关于连续函数空间 $C(X)$ 的 Asplund 性质的综述
A survey on the Asplund property for spaces $C(X)$ of continuous functions
- Institute of Mathematics, Czech Academy of Sciences（捷克科学院数学研究所）
- Faculty of Mathematics and Informatics, A. Mickiewicz University（亚当·密茨凯维奇大学数学与信息学院）
- Department of Mathematics, Ben-Gurion University of the Negev（内盖夫本-古里安大学数学系）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本综述为局部凸空间中的 Asplund 性质建立统一框架，完全刻画了连续函数空间 $C_k(X)$ 的 Asplund 性质，并用纯拓扑方法重新证明了经典结果。
AI中文摘要：
Asplund 性质在巴拿赫空间理论中扮演着重要角色，因为它与连续凸函数的可微性、优化问题以及巴拿赫空间的弱拓扑有关。鉴于文献中针对局部凸空间提出的多种不等价定义，本综述为在巴拿赫空间框架之外研究 Asplund 性质提供了一个统一框架。我们研究的主要对象是赋予紧开拓扑的连续函数局部凸空间 $C_k(X)$，其中 $X$ 是任意 Tychonoff 空间。我们完全刻画了 $C_k(X)$ 的 Asplund 性质，用底层空间 $X$ 的拓扑性质来表征。作为关键步骤，我们重新审视了几个经典结果的证明，包括 Namioka--Phelps 定理。我们的方法不依赖于可微性技术，仅依靠拓扑方法。论述是自包含的，所有主要结果都提供了完整证明，使论文对专家和新手都易于理解。
英文摘要：
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.
