CONTENTS Introduction................................................................................................................................5 Preliminaries..............................................................................................................................5 A. Vector lattices......................................................................................................................6 B. Measure theory.................................................................................................................12 Musielak-Orlicz spaces............................................................................................................15 1. Some properties of Musielak-Orlicz spaces.......................................................................15 2. Isomorphisms between Musielak-Orlicz spaces.................................................................20 3. Drewnowski-Orlicz theorem...............................................................................................23 Representations of Orlicz lattices............................................................................................24 4. Basic properties of orthogonal additive modulars and examples of Orlicz lattices.............24 5. The Main Representation Theorem for Orlicz lattices........................................................32 6. Representation of Orlicz lattices by Orlicz spaces.............................................................40 7. Ultraproducts of some Orlicz lattices..................................................................................48 Notes and comments...............................................................................................................57 References..............................................................................................................................61
Modifying ideas presented in [14] we prove that a complete metrizable locally solid Riesz space \(E\) contains a linear subspace linearly homeomorphic to \(\mathbb R^{\mathbb N}\) iff \(E\) contains a sublattice order isomorphic to \(\mathbb R^{\mathbb N}\).
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We investigate relationships joining the order continuity of a norm in a Banach lattice and some composition properties of L-weakly and M-weakly compact operators. Our results improve Proposition 3.6.16 from [5].
We describe, using elementary methods, the K\"othe dual of variable Lebesgue spaces \(L^{p(\cdot)},\) called also Nakano spaces, independenly for \(p(\cdot) \in (1, \infty)\) and \(p(\cdot) \in (0, 1)\). The case when \(p(\cdot) \in [1, \infty]\) is also included.
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