\(um\)-topology in multi-normed vector lattices (Q1753002)

Let \(M = \{m_\lambda\}_{\lambda\in\Lambda}\) be a separating family of lattice seminorms on a vector lattice \(X\), then \((X,M)\) is a multi-normed vector lattice (or MNVL). A net \(x_{\alpha}\) in \((X,M)\) \(m\)-converges to \(x\) iff \({m_\lambda}(x_\alpha-x) \to 0\) for all \(\lambda\in\Lambda\). A net \(x_{\alpha}\) in \((X,M)\) unboundedly converges to \(x\) iff \(|x_\alpha-x|\wedge u\) \(m\)-converges to \(0\) for all positive \(u\in X\). This concept generalizes \(un\)-convergence and \(uaw\)-convergence, while specializing \(up\)-convergence. The authors show that \(um\)-convergence is that of the so-called \(um\)-topology. In the case of an \(m\)-complete metrizable \((X,M)\), the \(um\)-topology is metrizable iff \(X\) has a countable topological orthogonal system. Using \(um\)-completeness, the authors characterize MNVLs with both Lebesgue and Levi properties. Some other applications are considered.