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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.
##### MSC:
 4.6e+31 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)