Some convexity properties of Musielak-Orlicz spaces of Bochner type.(English)Zbl 0609.46015

It is shown here that if a Banach space X and Musielak-Orlicz space $$L_{\phi}$$ are both locally uniformly rotund or uniformly rotund in every direction then the space $$L_{\phi}(X)$$ of Bochner type has the same properties. Moreover criteria for these properties have been given for a subspace of finite elements $$E_{\phi}(X)$$.

MSC:

 46E40 Spaces of vector- and operator-valued functions 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46B25 Classical Banach spaces in the general theory