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Mixed topology on normed function spaces. I. (English) Zbl 0756.46005
Summary: In the theory of two-norm spaces the notions of \(\gamma\)-convergence and \(\gamma\)-linear functional are of importance. In every normed function space (on a \(\sigma\)-finite measure space) \(\gamma\)-convergence can be defined in a natural way. In this paper we study locally solid topologies on a normed function space satisfying the continuity property with respect to the \(\gamma\)-convergence. We call such topologies “uniformly Lebesgue”. These investigations are closely related to Wiweger’s theory of mixed topologies. Namely, the appropriate mixed topology on a normed function space is the finest uniformly Lebesgue topology. We give a characterization of \(\gamma\)-linear functionals on a normed function space, and we show that \(\gamma\)-linear functionals have the extension property. Applications to \(L^ p\)-spaces and to Orlicz, Lorentz and Marcinkiewicz spaces are given. [For part II see the review below].

46A70 Saks spaces and their duals (strict topologies, mixed topologies, two-norm spaces, co-Saks spaces, etc.)
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Zbl 0756.46006