Characterizations of some monotonicity properties of a lattice norm in Musielak-Orlicz spaces. (English) Zbl 0714.46027

Kurc, W.

Acta Univ. Carol., Math. Phys. 30, No. 2, 91-94 (1989).

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

Keywords:

Banach lattice; monotone norm; uniformly monotone; UM Banach lattice; KB space; order dual of the Banach lattice; Musielak-Orlicz spaces