On some proximinal subspaces of modular spaces
Let \(X\) be a vector lattice, let \(\rho :X\rightarrow \left[ 0,\infty \right] \) be a semimodular and let \[ X_{\rho }=\left\{ x\in X:\lim\nolimits_{x\rightarrow 0}\rho \left( \lambda x\right) =0\right\} \] be the associated modular space and let \(S\) be an ideal in \(X_{\rho }.\) Equip the modular space \(X_{\rho }\) with the distance \(d\left( x,y\right)=\left|x-y\right|_{\rho }=\inf \{ \lambda >0:\rho \left( \frac{x-y}{\lambda }\right) \leq \lambda \} .\) If \(\rho \) is convex then on \(X_{\rho }\) one gives also the distance \(d_{L}\) and \(d_{0}\) generated by the Luxemburg norm \(\left\|x\right\|_{L}=\inf \left\{ \lambda >0,\rho \left( \frac{x}{\lambda }\right) \leq 1\right\} \) and by the Amemiya-Orlicz norm, \(\left\|x\right\|_{0}=\inf \left\{ \frac{1}{k}\left( 1+\rho \left( kx\right) \right) :k>0\right\} ,\) respectively. The authors obtain necessary and sufficient conditions in order that the subspace \(H\left( S\right) =\{x\in X_{\rho }:\forall \lambda >0,\exists s\in S,\rho \left( \frac{x-s}{\lambda }\right) <\infty \} \) be proximinal with respect to the distance \(d,d_{L}\) and \(d_{0}\).
scientific article; zbMATH DE number 1446781