Modular topologies on vector spaces
Mohamed Khamsi, Jan Lang, Osvaldo Mendez
TL;DR
This work investigates topologies induced by convex modulars on vector spaces when the $Δ_2$-property fails, with a focus on variable exponent spaces $\ell^{(p_n)}$ and $L^{p(\cdot)}$ and their relevance to unbounded $p(x)$-Laplacian problems. It develops the modular topology $\tau_{\rho}$ and the family of modular topologies $\tau_{\lambda}$, clarifying their connections to the Luxemburg norm and modular convergence, and identifies when modular balls are open (tied to bounded exponents and $Δ_2$). The authors show that, in general, modular topology can differ from norm topology in the absence of $Δ_2$, and modular duality may diverge from Banach duality, presenting implications for duality theory in non-linear, non-$Δ_2$ settings. Finally, they apply modular concepts to variational problems with nonstandard growth, demonstrating the natural modular closure $V^{1,p(\cdot)}_0(\Omega)$ yields a unique minimizer for the Dirichlet energy when $p(x)$ is unbounded, underscoring the practical significance for PDEs with variable exponents.
Abstract
This paper addresses the topological structures induced on vector spaces by convex modulars that do not satisfy the $Δ_2$ 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 \(Δ_2\)-condition, and duality with respect to modular topologies.