Orlicz lattices with modular topology. I. (English) Zbl 0679.46021

Nowak, Marian

Commentat. Math. Univ. Carol. 30, No. 2, 261-270 (1989).

For a given Riesz space X and a modular \(\rho\), the author has considered Orlicz lattices (X,\(\rho)\) with a modular topology \(\tau^{{\hat{\;}}}_{\rho}\). The main results are theorems which contain properties of \(\tau^{{\hat{\;}}}_{\rho}\) and linear functionals f on X (Theorems 2.1-2.6) and theorems on \(\tau^{{\hat{\;}}}_{\rho}\) and f in case when X is an ideal of the set of all real valued measurable and finite a.e. functions (Theorems 3.1 and 3.2).
The last part of the paper contains very interesting examples of Orlicz lattices, with modular topology and moreover remarks on \((L^{\infty},\rho_{\infty})\), \(\rho_{\infty}\) and mixed topology \(\gamma (\tau_{\infty},\tau_ 0/L^{\infty})\). The properties: \(\sigma\)-Lebesgue, \(\sigma\)-Fatou and \(\sigma\)-Levi are very important in that theory.

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

Keywords:

Riesz space; modular; Orlicz lattices; modular topology; mixed topology; \(\sigma\)-Lebesgue; \(\sigma\)-Fatou; \(\sigma\)-Levi

Citations:

Zbl 0679.46022