2015/09/25 by Niushan Gao, Gao, Niushan, Vladimir G. Troitsky +3 · 7 citations
Computer Science · Mathematics · #Advanced Banach Space Theory #Approximation Theory and Sequence Spaces #FOS: Mathematics #Functional Analysis (math.FA) #Optimization and Variational Analysis
paper · pdf · doi:10.48550/arxiv.1509.07914
openalex publication_date 2015/09/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A net (xα) in a vector lattice X is said to uo-converge to x if |xα-x|\wedge u\xrightarrow\rm o0 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].