Bourgain-uo sequential completeness in vector lattices
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915873321648128 |
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| author | Kania, Tomasz Swaczyna, Jarosław |
| author_facet | Kania, Tomasz Swaczyna, Jarosław |
| contents | We revisit Bourgain's 1981 counterexample to the sequential completeness of the `pointwise plus domination' convergence on $\ell_1$ 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 $(x_n)$ in a vector lattice $E$ is called Buo-Cauchy if for every strictly increasing sequence $(n_k)$ the differences $x_{n_{k+1}}-x_{n_k}$ 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 $\mathrm{FBL}(E)$ are not sequentially Buo-complete whenever $\dim E>1$.
On the positive side, we prove that the classical sequence lattices $c_0$ and $\ell_\infty$ 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 $\mathrm{Lip}_b(X)$ of bounded Lipschitz functions on a metric space $(X,d)$ is sequentially Buo-complete if and only if $X$ is uniformly discrete. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_14949 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bourgain-uo sequential completeness in vector lattices Kania, Tomasz Swaczyna, Jarosław Functional Analysis 46B42, 46B45, 46E05, 46E30 We revisit Bourgain's 1981 counterexample to the sequential completeness of the `pointwise plus domination' convergence on $\ell_1$ 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 $(x_n)$ in a vector lattice $E$ is called Buo-Cauchy if for every strictly increasing sequence $(n_k)$ the differences $x_{n_{k+1}}-x_{n_k}$ 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 $\mathrm{FBL}(E)$ are not sequentially Buo-complete whenever $\dim E>1$. On the positive side, we prove that the classical sequence lattices $c_0$ and $\ell_\infty$ 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 $\mathrm{Lip}_b(X)$ of bounded Lipschitz functions on a metric space $(X,d)$ is sequentially Buo-complete if and only if $X$ is uniformly discrete. |
| title | Bourgain-uo sequential completeness in vector lattices |
| topic | Functional Analysis 46B42, 46B45, 46E05, 46E30 |
| url | https://arxiv.org/abs/2512.14949 |