标题： 向量空间上的模拓扑结构 显示英文标题

标题： Modular topologies on vector spaces

摘要： 本文研究了由满足凸模条件但不满足 $\Delta_2$ 条件的凸模在向量空间上诱导出的拓扑结构，并特别关注其在可变指数空间（如 \( \ell^{(p_n)} \) 和 \( L^{p(\cdot)} \)）中的应用。 这项研究的动机在于其对涉及可变指数 $p(x)$-Laplacian 的边值问题的研究适用性，特别是在当 $p(x)$ 无界时，这是作者近期开辟的一个研究方向。 分析了基本的拓扑性质，包括分离公理、可数公理以及模收敛与连续性等经典拓扑概念之间的关系。 重点讨论了模拓扑与范数拓扑之间的关系，特别强调了模球的开性、\(\Delta_2\)-条件的影响以及模拓扑下的对偶性。 显示英文摘要

摘要： This paper addresses the topological structures induced on vector spaces by convex modulars that do not satisfy the $\Delta_2$ condition, with particular focus on their applications to variable exponent spaces such as \( \ell^{(p_n)} \) and \( L^{p(\cdot)} \). The motivation behind this investigation is its applicability to the study of boundary value problems involving the variable exponent $p(x)$-Laplacian when $p(x)$ is unbounded, a line of research recently opened by the authors. Fundamental topological properties are analyzed, including separation axioms, countability axioms, and the relationship between modular convergence and classical topological concepts such as continuity. Attention is given to the relation between modular and norm topologies. Special emphasis is placed on the openness of modular balls, the impact of the \(\Delta_2\)-condition, and duality with respect to modular topologies.