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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 \(\Delta_ 2\).

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