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Uniformly \(\mu \)-continuous topologies on Köthe-Bochner spaces and Orlicz-Bochner spaces. (English) Zbl 0970.46015
The author gives an account of locally solid topologies and studies mutual relations of uniformly \(\mu \)-continuous topologies, uniformly \(\mu \)-continuous solid pseudonorms, and uniformly summable pseudonorms. In Köthe-Bochner spaces he proves a characterization of uniformly \(\mu \)-continuous locally solid topologies in terms of uniformly summable pseudonorms.
Further, he shows that the finest uniformly \(\mu \)-continuous topology on an Orlicz-Bochner space is the generalized mixed topology, which coincides here with the concept of a strictly inductive limit of balanced topological spaces. In this frame a characterization of \(\gamma _{\varphi }\)-linear mappings is given.
MSC:
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46E40 Spaces of vector- and operator-valued functions
46A70 Saks spaces and their duals (strict topologies, mixed topologies, two-norm spaces, co-Saks spaces, etc.)
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