Complex extreme points and complex rotundity in Orlicz function spaces equipped with the \(p\)-Amemiya norm. (English) Zbl 1210.46013
Necessary and sufficient conditions in order that a point \(x\) of the unit ball \(B(L_{\Phi,p})\) of the complex Orlicz function space \(L_{\Phi,p}\) equipped with the \(p\)-Amemiya norm (\(1\leq p<\infty\)) is a complex extreme point are given. This condition says that \(\mu\{t\in T\colon k\left|x(t)\right|<a_\Phi\}=0\) for any \(k\in K_p(x)\), where \(a_\Phi=\sup\{u\geq 0\colon\Phi(u)=0\}\) and \(K_p (x)\) is the set of all positive numbers \(k\) such that the infimum in the definition of the \(p\)-Amemiya norm of \(x\) is attained at \(k\).
As a consequence of this result, it is proved that the space \(L_{\Phi,p}\) is complex strictly convex if and only if \(a_\Phi=0\). Criteria for complex extreme points of \(B(L_{\Phi,\infty})\) and complex strict convexity of the space \(L_{\Phi,\infty}\) are also presented.
As a consequence of this result, it is proved that the space \(L_{\Phi,p}\) is complex strictly convex if and only if \(a_\Phi=0\). Criteria for complex extreme points of \(B(L_{\Phi,\infty})\) and complex strict convexity of the space \(L_{\Phi,\infty}\) are also presented.
Reviewer: Henryk Hudzik (Poznań)
MSC:
| 46B20 | Geometry and structure of normed linear spaces |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
Keywords:
Orlicz function spaces; complex extreme points; complex strict convexity; \(p\)-Amemiya normReferences:
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