On Musielak-Orlicz Sequence Spaces with an Asymptotic ℓ∞ dual

Abstract

We investigate Mushielak-Orlicz sequence spaces ${\ell }_{\mathrm{\Phi }}$ with a dual ${\ell }_{\mathrm{\Phi }}^{\ast }$, which is stabilized asymptotic ${\ell }_{\mathrm{\infty }}$ with respect to the unit vector basis. We give a complete characterization of the bounded relatively weakly compact subsets $K\subset {\ell }_{\mathrm{\Phi }}$. We prove that ${\ell }_{\mathrm{\Phi }}$ is saturated with asymptotically isometric copies of ${\ell }_1$ and thus ${\ell }_{\mathrm{\Phi }}$ fails the fixed point property for closed, bounded convex sets and non--expansive (or contractive) maps on them.