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A generalized mixed topology on Orlicz spaces. (English) Zbl 0806.46032
Summary: Let $$L^ \varphi$$ be an Orlicz space defined by an arbitrary Orlicz function $$\varphi$$ over a positive measure space $$(\Omega,\Sigma,\mu)$$ and provided with its usual $$F$$-norm $$\|\cdot\|_ \varphi$$. In $$L^ \varphi$$ a natural convergence can be defined as follows: a sequence $$(x_ n)$$ in $$L^ \varphi$$ is said to be $$\gamma_ \varphi$$- convergent to $$x\in L^ \varphi$$ whenever $$x_ n\to x$$ $$(\mu-\Omega)$$ and $$\sup\| x_ n\|_ \varphi< \infty$$. In this paper we examine some kind of generalized inductive-limit topology (in the sense of Turpin) $${\mathcal I}^ \varphi_ I$$ in $$L^ \varphi$$ that generates our $$\gamma_ 0$$-convergence in $$L^ \varphi$$. The main aim of the paper is to obtain a description of the topology $${\mathcal I}^ \varphi_ I$$ in terms of some family of $$F$$-norms defined by other Orlicz functions. As an application we obtain a topological characterization of the $$\gamma_ \varphi$$-convergence in $$L^ \varphi$$.

##### MSC:
 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46A70 Saks spaces and their duals (strict topologies, mixed topologies, two-norm spaces, co-Saks spaces, etc.)
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