Normal structure and weakly normal structure of Orlicz spaces. (English) Zbl 0760.46023
Let \(L_ M\) be an Orlicz space equipped with the Luxemburg norm; T. Landes [Trans. Am. Math. Soc. 285, 523-534 (1984; Zbl 0594.46010)] has given necessary and sufficient conditions under which \(L_ M\) has normal structure.
In this paper the authors study the case when \(L_ M\) is equipped with the Orlicz norm. The result is that \(L_ M\) has normal structure iff the \(N\)-function \(M\) satisfies:
there exist \(a>0\), \(C>1\) such that, for any \(u>A\) the function \(M\) is not affine on \([u,Cu]\).
In this paper the authors study the case when \(L_ M\) is equipped with the Orlicz norm. The result is that \(L_ M\) has normal structure iff the \(N\)-function \(M\) satisfies:
there exist \(a>0\), \(C>1\) such that, for any \(u>A\) the function \(M\) is not affine on \([u,Cu]\).
Reviewer: P.Mazet (Paris)
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 |
| 46B25 | Classical Banach spaces in the general theory |
