Orlicz lattices with modular topology. II. (English) Zbl 0679.46022

Nowak, Marian

Commentat. Math. Univ. Carol. 30, No. 2, 271-279 (1989).

(X,\(\rho)\) denotes an Orlicz lattice, where X is a Riesz space, \(\rho\) is a convex modular and \(\tau^{{\hat{\;}}}_{\rho}\) denotes a modular topology. In the first part of the note there are given very interesting properties of the space \((X,\tau^{{\hat{\;}}}_{\rho})\). In the following we have a remark on the associated space and a theorem which says that sequentially modular continuous functionals on Orlicz lattices in \(L^ 0\) have the extension property. Very important are theorems given in the part 4, where the modular topology \(\tau^{{\hat{\;}}}_{\rho}\) on Orlicz spaces \(L^{\phi}\) is considered. The last part of the note contains the various relations between different spaces of linear mappings (between Orlicz lattices). [For part I see the review above.]

Reviewer: A.Waszak

MSC:

46E30	Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46A40	Ordered topological linear spaces, vector lattices
46A22	Theorems of Hahn-Banach type; extension and lifting of functionals and operators

Keywords:

Riesz space; convex modular; modular topology; associated space; sequentially modular continuous functionals on Orlicz lattices; extension property; spaces of linear mappings

Citations:

Zbl 0679.46021