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Thursday, 28 May 2026 - 15:00 to 16:30 <br />
Place: IM, konÃrna
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Speaker: Jan Lang, Ohio State University, USA<br />
Title: Notes about modular-based topologies
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Abstract <br />
<p>This talk concerns the topology generated by modular convergence in vector spaces equipped with a convex modular $\rho$, with particular emphasis on the case where $\rho$ does not satisfy the $\Delta_2$-condition. We show that the modular topology is a topological vector space topology precisely when $\rho$ satisfies $\Delta_2$. In the absence of this condition, several familiar features of normed spaces break down: modular balls need not be open, they may have empty interior, and modular convergence may be strictly weaker than Luxemburg norm convergence.</p><p>The general theory is illustrated in the variable exponent spaces $\ell^{(p_n)}$ and $L^{p(.)}(\Omega)$, where unbounded exponents lead to genuinely non-normable modular phenomena. We also discuss applications to Dirichlet energy minimization and weak solutions of boundary value problems for the p(x)-Laplacian with unbounded exponent.</p>
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For more information see the seminar web page at <br />
https://www.math.cas.cz/index.php/events/seminar/6
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Set Theory and Analysis mailing list <br />
settfa@math.cas.cz <br />
https://list.math.cas.cz/listinfo/settfa@math.cas.cz
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