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On a class of universal Orlicz function spaces. (English) Zbl 0799.46030
J. Lindenstrauss and L. Tzafriri [Isr. J. Math. 11, 355–379 (1972; Zbl 0237.46034)] have given examples of universal Orlicz sequence space \(l^\Psi\) for every Orlicz sequence spaces \(l^\varphi\) with \(\varphi\) an Orlicz function \(c\)-convex and \(d\)-concave for every \(1 \leq c < d < \infty\) and it was proved that every space \(l^\varphi\) is isomorphic to a complemented subspace of \(l^\Psi\).
In this paper the author gives a class of universal Orlicz function spaces \(L^\Psi [0,1]\), which are universal for a prefixed class of Orlicz sequence spaces \(l^\varphi\) and the spaces \(l^\varphi\) are also isomorphic to complemented subspaces of \(L^\Psi [0,1]\).
As a consequence the author shows that every separable Nakano sequence space \(l^{(p_ n)}\) can be isomorphically represented as a weighted Orlicz sequence space \(l^\Psi (w)\) for a suitable Orlicz function \(\Psi\) and some summable weight sequence \(w\).
MSC:
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46A45 Sequence spaces (including Köthe sequence spaces)
46B45 Banach sequence spaces
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