Uniform monotonicity of Orlicz spaces equipped with the Mazur–Orlicz F‐norm and dominated best approximation in F‐normed Köthe spaces

Abstract

We collect several useful equivalent conditions for order continuity in F-normed Köthe spaces. Next, we study uniform monotonicity of Orlicz function (sequence) spaces equipped with the Mazur-Orlicz F-norm. We show necessary and sufficient conditions, separately. Moreover, we study the best dominated approximation problems in F-normed Köthe spaces showing that order continuity and strict monotonicity are crucial tools in these problems. Finally, we apply the general results in Orlicz spaces. We focus mainly on these methods and techniques which are essentially different and more delicate in comparison to the case of normed Köthe spaces.