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Smooth points of Orlicz spaces equipped with Luxemburg norm. (English) Zbl 0795.46016

Let \(\Phi\) be an Orlicz function on \((-\infty,\infty)\) and \(\mu\) be a non-atomic or a counting measure on a set \(T\). Recall, that the Orlicz space \(L^ \Phi (\mu)\) coincides with the set \(\{x(t)\): there exists \(\lambda>0\) such that \(\int_ T\Phi ({{x(t)}\over \lambda})d\mu< +\infty\}\). The Luxemburg norm on \(L^ \Phi(\mu)\) is defined by the formula \(\| x\|_ \Phi= \inf\{\lambda>0\): \(\int_ T \Phi ({{x(t)} \over \lambda}) d\mu\leq 1\}\), and is equivalent to the usual Orlicz norm. An element \(0\neq x\in L^ \Phi (\mu)\) is said to be smooth if there exists a unique functional \(x^*\in (L^ \Phi (\mu))^*\) such that \(x^*(x)= \| x\|_ \Phi\). The following result is proved.
Theorem 8. Let \(\mu\) be a non-atomic measure and \(\Phi\) be an Orlicz function which has only finite values. A point \(x\), \(\| x\|_ \Phi=1\), is smooth if and only if the following two conditions are valid:
(i) \(\int_ T \Phi ({{x(t)} \over \lambda}) d\mu< +\infty\) for some \(\lambda\), \(0<\lambda<1\);
(ii) \(\Phi\) is smooth at the point \(x(t)\) for \(\mu\)-a.e. \(t\in T\).
The case of counting measure is considered as well.

MSC:

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