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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 \(L^ \varphi\).

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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