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On uniformly convexifiable and B-convex Musielak-Orlicz spaces. (English) Zbl 0588.46022
It is proved in the case of a non-atomic measure as well as of a purely atomic measure that for Musielak-Orlicz spaces reflexivity, super- reflexivity and B-convexity are equivalent properties. This is proved basing on two key lemmas 1.1.2 and 1.2.4, which say that for any Musielak-Orlicz function $$\Phi$$ satisfying condition $$\Delta_ 2$$ (respectively condition $$\delta^ 0_ 2$$ in the case of a purely atomic measure) there exists a Musielak-Orlicz function $$\Phi_ 1$$ equivalent to $$\Phi$$ and satisfying uniform condition $$\Delta_ 2$$ (respectively uniform condition $$\delta_ 2)$$. Moreover, some other lemmas are given.

##### MSC:
 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46B20 Geometry and structure of normed linear spaces 46B10 Duality and reflexivity in normed linear and Banach spaces