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Keywords:
extreme points; vector valued continuous functions; compact linear operators; Orlicz spaces
extreme points; vector valued continuous functions; compact linear operators; Orlicz spaces
Summary:
Let $E^{\varphi }(\mu )$ be the subspace of finite elements of an Orlicz space endowed with the Luxemburg norm. The main theorem says that a compact linear operator $T:E^{\varphi }(\mu )\rightarrow C(\Omega )$ is extreme if and only if $T^{\ast }\omega \in \operatorname{Ext}\, B((E^{\varphi }(\mu ))^{\ast })$ on a dense subset of $\Omega $, where $\Omega $ is a compact Hausdorff topological space and $\langle T^{\ast } \omega ,x\rangle=(T x)(\omega )$. This is done via the description of the extreme points of the space of continuous functions $C(\Omega ,L^{\varphi }(\mu ))$, $L^{\varphi }(\mu )$ being an Orlicz space equipped with the Orlicz norm (conjugate to the Luxemburg one). There is also given a theorem on closedness of the set of extreme points of the unit ball with respect to the Orlicz norm.
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