The uniform Kadec–Klee property for Orlicz–Lorentz spaces

Abstract

We give sufficient conditions, as well as some necessary conditions, for the Orlicz–Lorentz space Λϕ,ω to have the weak-star uniform Kadec–Klee property. These results generalize the characterization of the weak-star uniform Kadec–Klee property in the Lorentz space Λω = Lω,1 due to Dilworth and Hsu.

Convex Komlós sets in Banach function spaces

“…Other examples of such spaces are Lorentz, Orlicz and Orlicz-Lorentz spaces. For more information about Banach function spaces see, for example, [1,9,4].…”

Rotundity and uniform rotundity of Orlicz‐Lorentz sequence spaces equipped with the Orlicz norm

“…Under the Luxemburg norm given by \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Vert x\Vert =\inf \lbrace \varepsilon >0:\rho _\varphi (x/\varepsilon )\le 1\rbrace$\end{document}, λ φ, ω is a Banach space (see 15). For the properties of λ φ, ω , which are similar to those of Λ φ, ω , one can see 1, 5, 6, 11, 12, 15–18.…”