Uo-convergence and its applications to Cesàro means in Banach lattices

Abstract

Abstract. A net (x α ) in a vector lattice X is said to uo-converge to x if |x α −x|∧u o − → 0 for every u ≥ 0. In the first part of this paper, we study some functional-analytic aspects of uo-convergence. We prove that uo-convergence is stable under passing to and from regular sublattices. This fact leads to numerous applications presented throughout the paper. In particular, it allows us to improve several results in [26,27]. In the second part, we use uo-convergence to study convergence of Cesàro means in Banach lattices. In particular, we establish an intrinsic version of Komlós' Theorem, which extends the main results of [35,16,31] in a uniform way. We also develop a new and unified approach to Banach-Saks properties and Banach-Saks operators based on uo-convergence. This approach yields, in particular, short direct proofs of several results in [21,24,25].

Unbounded norm topology beyond normed lattices

“…We say that y α uo-converges to y and write y α uo − → y if |y α − y| ∧ u o − → 0 for every u ∈ Y + . We refer the reader to [GTX17] for a review of order and uo-convergence. We will only mention three facts here.…”

“…Recall that a sublattice Z of Y is regular if z α ↓ 0 in Z implies z α ↓ 0 in Y . In this case, Theorem 3.2 of [GTX17] asserts that z α…”

Nonstandard hulls of lattice-normed ordered vector spaces

“…We continue with further basic notions in LNOVSs, which are motivated by their analogies for vector lattices and for LNVLs (see, for example, [4][5][6]19,20,[25][26][27]). …”

Completeness of unbounded convergences

“…The section starts with a counterexample to a question posed in [LC], and culminates in a proof that a Banach lattice is (sequentially) boundedly uo-complete iff it is (sequentially) monotonically complete. This gives the final solution to a problem that has been investigated in [Gao14], [GX14], [GTX17], and [GLX].…”

“…We write x α uo − → x and say that (x α ) uo-converges to x ∈ X if |x α − x| ∧ u o − → 0 for every u ∈ X + . For facts on uo-convergence, the reader is referred to [GTX17]. In particular, [GTX17, Theorem 3.2] will be used freely.…”