# zbMATH — the first resource for mathematics

On some equivalent conditions in Musielak-Orlicz spaces. (English) Zbl 0564.46022
Let $$L_ M^{\mu}$$ denote the Musielak-Orlicz space of vector-valued functions, over a space of atomless measure $$\mu$$, equipped with the Luxemburg norm N. It is proved under suitable assumptions that the following assertions are equivalent:
(a) The N-function (i.e. Musielak-Orlicz function) M satisfies condition $$\Delta_ 2$$, i.e. there are a constant $$K>0$$, a set $$T_ 0$$ of measure zero and a non-negative $$\mu$$-summable function h such that $$M(t,2x)\leq KM(t,x)+h(t)$$ for all $$t\in T\setminus T_ 0$$ and all $$x\in X.$$
(b) $$\int_{T}M(t,f(t)/N(f))d\mu =1$$ for $$0\neq f\in L_ M^{\mu}.$$
(c) There do not exist norm-one functions f and g with disjoint supports such that $$N(f+g)=1.$$
(d) No closed subspace of $$L_ M^{\mu}$$ is isometric to $$\ell^{\infty}.$$
(e) No closed subspace of $$L_ M^{\mu}$$ is isomorphic to $$\ell^{\infty}.$$
(f) No closed subspace of $$L_ M^{\mu}$$ is isomorphic to $$c_ 0.$$
It is proved also in the case of finite-dimensional Banach space X that $$L_ M^{\mu}$$ is reflexive iff M and its complementary function $$M^*$$ both satisfy condition $$\Delta_ 2$$.

##### MSC:
 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46E40 Spaces of vector- and operator-valued functions 46B25 Classical Banach spaces in the general theory