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On two linear topologies on Orlicz spaces \(L^{*\phi}\). I. (English) Zbl 0602.46020
Let \(\phi\) be a \(\phi\)-function and let E be a finite-dimensional Euclidean space with the usual Lebesgue measure. In this paper there are considered two linear topologies \({\mathcal T}^{\triangleleft \phi}\) and \({\mathcal T}^{\ll \phi}\) on the Orlicz space \(L_ E^{*\phi}\), generated by the families of F-norms \(\{\| \cdot \|_{\psi}:\psi \triangleleft \phi \}\) and \(\{\| \cdot \|_{\psi}:\psi \ll \phi \}\) respectively, where \(\psi \triangleleft \phi\) means that lim sup \(\psi\) (cu)/\(\phi\) (u)\(<\infty\) as \(u\to 0\) and \(u\to \infty\) for all \(c>0\) and \(\psi \ll \phi\) means that lim \(\psi\) (cu)/\(\phi\) (u)\(=0\) as \(u\to 0\) and \(u\to \infty\) for all \(c>0\). The sequential convergence in the topologies \({\mathcal T}^{\triangleleft \phi}\) and \({\mathcal T}^{\ll \phi}\) is compared with the modular convergence in \(L_ E^{*\phi}\). It is proved that \(L_ E^{*\phi}\) with \({\mathcal T}^{\triangleleft \phi}\) and \({\mathcal T}^{\ll \phi}\) is complete and separable, and moreover, for \(\phi\) being an N-function, locally convex.
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)