Modular Topologies on Vector Spaces: Structure and Normability

Abstract

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.