AI 中文总结
本文研究向量空间上的模拓扑，证明其与空间结构相容等价于模满足Δ₂条件，刻画了Luxemburg范数拓扑的性质，并将理论应用于变指数空间求解开放边值问题。
AI 中文摘要
我们研究在配备凸模的向量空间上，由模收敛生成的拓扑。尽管模收敛长期以来在模函数空间理论中占据核心地位，但据作者所知，其诱导的拓扑此前从未被作为独立的数学对象加以研究。我们的主要结果表明，模拓扑与向量空间结构相容当且仅当基础模满足Δ₂条件。因此，Δ₂条件可获得纯拓扑刻画。我们还证明，Luxemburg范数拓扑是包含所有缩放模拓扑的最弱第一可数拓扑。我们将该理论应用于变指数序列空间及超越经典Δ₂理论的Lebesgue空间，并将所得结果用于求解指数p无界的W^{1,p(x)}中Dirichlet积分的极小化问题，从而解决了一个开放的边值问题。
英文摘要
We investigate the topology generated by modular convergence on vector spaces endowed with a convex modular. Although modular convergence has long played a central role in the theory of modular function spaces, the topology it induces has, to the authors' best knowledge, not previously been studied as an independent mathematical object. Our principal result establishes that the modular topology is compatible with the vector space structure if and only if the underlying modular satisfies the $Δ_2$-condition. Consequently, the $Δ_2$-condition admits a purely topological characterization. We also show that the Luxemburg norm topology is the weakest first-countable topology containing all scaled modular topologies. We apply the theory to variable exponent sequence and Lebesgue spaces beyond the classical $Δ_2$-theory and present an application of our results to the minimization of the Dirichlet integral in $W^{1,p(x)}$ with unbounded exponent $p$, which allows us to solve an open boundary value problem.
