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Smoothness of Orlicz spaces. (English) Zbl 0641.46010
From author’s introduction: A Banach space X is said to be smooth at the point \(x\in X\) if there exists a unique \(f_ x\in X^*\) such that \(f_ x(x)=\| f_ x\| \| x\| =\| x\|\). The mapping \(x\to f_ x\) is called a support mapping. X is said to be smooth if it is smooth at every point in its unit sphere S(X). X said to be very (strong or uniformly) smooth if it is smooth and its support mapping is norm to weak (norm or norm uniformly, respectively) continuous from S(X) to \(S(X^*).\)
This paper deals with smoothness of Orlicz spaces equipped with Orlicz norm. Criteria for the smoothness, very smoothness, strong smoothness and uniform smoothness of these spaces are given.
Reviewer: A.A.Mekler

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