Problems of existence of order copies of $$\ell ^\infty $$ ℓ ∞ and $$L_p(\nu )$$ L p ( ν ) in some non-Banach Köthe spaces

Abstract

For a monotone Orlicz function Φ taking only values 0 and ∞, it is showed that in both cases, the s-homogeneous norm • Φ,s , if Φ is s-convex (0 < s ≤ 1) and the Mazur-Orlicz F-norm • Φ , if Φ is non-decreasing on R + , we have that L Φ (μ) = L ∞ (μ) and both these norms are proportional to • ∞. The problems of existence of order linearly isometric copy of ∞ \B ∞ (0, ε) for any ε > 0 as well as an order linearly isometric copy of the whole ∞ in Orlicz F-normed function and sequence spaces are considered. In the last section the problem of the existence of order linearly isometric copies of L p (ν) with 0 < p ≤ 1 in F-normed Orlicz spaces are considered.

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

“…Clearly, the theory of normed Orlicz spaces equipped with the Luxemburg-Nakano norm is very well known. Recently, some authors studied 𝐹-normed Orlicz and Musielak-Orlicz spaces with the Mazur-Orlicz 𝐹-norm (see [10,13,14,18,31,32,34,35]). Moreover, the quasi-normed Calderón-Lozanovskiȋ spaces (in particular Orlicz spaces) have been studied in [19].…”