{
  "abstract": "We resolve three questions arising from the study of coordinate systems and order convergence in Banach lattices. Firstly, every order (respectively, $σ$-order) basis $(e_n)$ for a Banach lattice $X=[e_n]$ is automatically a Schauder basis. In fact, the associated partial sum projections satisfy a maximal inequality, and hence every such basis is uniform. Secondly, we show that there is a separable Banach lattice $X$ in which the set $$ X_{\\downarrow0}=\\{(x_n)\\in X^{\\mathbb N}:x_1\\geqslant x_2\\geqslant\\ldots\\geqslant 0=\\inf_nx_n\\} $$ is $\\mathbfΠ^1_1$-complete. Finally, we construct a separable Banach lattice $Y$ for which the set $Y^{\\uparrow\\infty}$ of increasing positive sequences which are not order bounded is $\\mathbfΠ^1_1$-complete. It follows that neither $X_{\\downarrow0}$ nor $Y^{\\uparrow\\infty}$ can be described in any simpler way than by the nonexistence of positive lower and upper bounds, respectively.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.33462",
  "author_total": 4,
  "authors": [
    {
      "name": "A. Avilés"
    },
    {
      "name": "C. Rosendal"
    },
    {
      "name": "M. A. Taylor"
    },
    {
      "name": "P. Tradacete"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/423xkgfnvd437.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.33462",
  "fulltext_url": null,
  "id": "423xkgfnvd437",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.33462",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7223991936",
  "pdf_url": "https://arxiv.org/pdf/2610.33462",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/423xkgfnvd437.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.33462",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.33462"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Coordinate systems and maximal order complexity in Banach lattices",
  "title_text": "Coordinate systems and maximal order complexity in Banach lattices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/423xkgfnvd437",
  "venue": "arXiv",
  "versions": []
}