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Uniform convexity of Musielak-Orlicz spaces with Luxemburg’s norm. (English) Zbl 0595.46027
Summary: We give some sufficient conditions in order that the Cartesian product \(L_ M^{\mu}(T,X)\times L_ N^{\phi}(S,Y)\) of Musielak-Orlicz spaces \(L_ M^{\mu}(T,X)\) and \(L_ N^{\phi}(S,Y)\) with Luxemburg’s norm be uniformly convex and we prove necessity of some of them. Next, we give some corollaries and some examples of \(\phi\)-functions with parameter, which generate uniformly convex Musielak-Orlicz spaces. These results are generalizations of the respective results of W. A. J. Luxemburg [Banach function spaces, Thesis, Delft (1955; Zbl 0068.092)] and H. Nakano [Topology and linear topological spaces (1951)].

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)