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Characterizations of some monotonicity properties of a lattice norm in Musielak-Orlicz spaces. (English) Zbl 0714.46027
Given a Banach lattice X with a monotone norm $$\| \cdot \|$$, $$(X,\| \cdot \|)$$ is said to be uniformly monotone (UM) if for every $$\epsilon >0$$ there holds $$\delta_+(\epsilon)=\inf_{U^+_{\epsilon}}(\| z\| -1)>0,$$ where $$U^+_{\epsilon}=\{z=x+y:\;x,y\geq 0,\quad \| x\| =1,\quad \| y\| \geq \epsilon \}.$$ Each UM Banach lattice is a KB space, i.e. $$j(X)=(X^*)^{\sim}_ n$$ under the evaluation $$j: X\to (X^*)^*,$$ where $$X^*$$ denotes the order dual of the Banach lattice X and $$X^{\sim}$$ consists of all order continuous functionals in X. The author shows that this implication is strict for Musielak-Orlicz spaces.
Reviewer: B.P.Duggal

##### MSC:
 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46B42 Banach lattices
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