2025/12/16 by Kania, Tomasz, Swaczyna, Jarosław
Computer Science · Mathematics · #46B42 #46B45 #46E05 #46E30 #Advanced Banach Space Theory #FOS: Mathematics #Fixed Point Theorems Analysis #Functional Analysis (math.FA) #Optimization and Variational Analysis
paper · doi:10.48550/arxiv.2512.14949
openalex publication_date 2025/12/16 · openalex created_date 2025/12/19 · openalex updated_date 2026/07/28
We revisit Bourgain's 1981 counterexample to the sequential completeness of the `pointwise plus domination' convergence on ℓ1 from the perspective of vector lattices. In this setting, we show that for sequences the associated notion of Bourgain--uo convergence coincides with ordinary order convergence. Motivated by Bourgain's construction, we introduce a strengthened, subsequence-invariant notion of Cauchy sequence: a sequence (xn) in a vector lattice E is called Buo-Cauchy if for every strictly increasing sequence (nk) the differences x_nk+1-xnk converge to 0 in order in E. We first show that sequential Buo-completeness forces σ-order completeness. Thus every non-σ-order complete vector lattice fails sequential \Buo-completeness. In particular, free Banach lattices FBL(E) are not sequentially Buo-complete whenever dim E>1. On the positive side, we prove that the classical sequence lattices c0 and ℓ_∞ are sequentially Buo-complete: every Buo-Cauchy sequence converges in order, and hence in the Buo sense. Finally, we obtain a sharp metric characterisation for bounded Lipschitz function lattices: the vector lattice Lipb(X) of bounded Lipschitz functions on a metric space (X,d) is sequentially Buo-complete if and only if X is uniformly discrete.