On some modes of convergence in spaces with the weak Banach-Saks property. (English) Zbl 1298.46028

Jakszto, Marian

Real Anal. Exch. 38(2012-2013), No. 2, 487-492 (2013).

A typical result of the paper is the following:
Theorem. Let \(F\) be a reflexive Banach function space (on a complete \(\sigma\)-finite measure space) with the weak Banach-Saks property and \((f_n)\) be a bounded sequence in \(F\) converging a.e. to \(f\), then \(f\in F\) and \((f_n)\) converges weakly to \(f\).

Reviewer: Hans Weber (Udine)

MSC:

46E30	Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
28A20	Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence

Keywords:

weak Banach-Saks property; weak convergence; convergence a.e.