Some fundamental properties for duals of Orlicz spaces
Let \(\Phi: \mathbb R\to [0,\infty]\) be an Orlicz function, i.e., \(\Phi\) is convex, even, left continuous on \([0,\infty)\), vanishes only at zero, and is not identically equal to infinity on \((0,\infty)\). Given a \(\sigma\)-finite non-atomic measure space \((G,\Sigma, \mu)\), let \(L^0(\mu)\) denote the space of real valued \(\mu\)-measurable functions defined on \(G\). Then the Orlicz space \(L_\Phi= \{x \in L^0(\mu): \exists c>0\;I_\Phi(cx) = \int_G \Phi(cx(t)) \,d\mu(t) < \infty\}\) is a Banach space equipped with the standard Luxemburg or Orlicz norm. The dual space \(L_\Phi^*\) is generated by the modular \(\rho^*(f)= I_\Psi(v) + \|\varphi\|\), where \(\Psi\) is the conjugate function to \(\Phi\), and \(f\in L_\Phi^*\) has a unique decomposition \(f = v + \varphi\) with \(v\in L_\Psi\) (which induces an integral functional) and \(\varphi\) is a singular functional. Several useful relations among \(I_\Phi\), \(\rho^*\) and the Luxemburg and Orlicz norms are presented. They are applied to characterize the extreme points of the dual space \(L_\Phi^*\) equipped with the Luxemburg norm. It is shown that a functional \(f = v + \varphi \in L_\Phi^*\) such that \(\|f\|=1\), \( 0\neq v \in L_\Psi, 0 \neq \varphi \in F\), where \(F\) is the set of singular functionals, is an extreme point if and only if {\parindent7.5mm \begin{itemize}\item[(i)] \(\rho^*(f) = 1\); \item[(ii)] \(\mu\{t\in G: v(t) \;\text{does not belong to the set of strict extreme points of}\;\Psi \} =0\); \item[(iii)] \(\varphi/\|\varphi\|\) is an extreme point of the unit ball of \(L_\Phi^*\). \end{itemize}}
scientific article; zbMATH DE number 5774951
scientific article