{
  "abstract": "We introduce and study the notion of \\emph{norming approximate orthogonality}, a two-parameter generalization of Birkhoff--James orthogonality in normed linear spaces. For $δ, \\varepsilon \\in [0,1)$ with $\\varepsilon < (1-δ)^2$, we say $x \\nperp y$ in $X$ if there exists $f \\in X^*$ with $|f(x)| \\geq (1-δ)\\|f\\|\\|x\\|$ and $|f(y)| \\leq \\frac{\\varepsilon}{1-δ}\\|f\\|\\|y\\|$, simultaneously relaxing both the norming condition on $x$ and the vanishing condition on $y$. It is proved that \\[ x\\nperp y \\iff \\|x+λy\\|\\geq (1-δ)\\|x\\|-\\frac{\\varepsilon}{1-δ}\\|λy\\|~\\qquad \\forall ~\\text{scalars}~λ. \\] This framework interpolates between two notions of approximate orthogonality in normed linear spaces due to Chmieliński and Dragomir, and recovers three existing notions of orthogonality in extreme cases: exact Birkhoff--James orthogonality at $δ= \\varepsilon = 0$, the approximate orthogonality of Chmieliński at $δ= 0$, and the approximate orthogonality of Dragomir at $\\varepsilon = 0$. A two-parameter proximity result generalizing Chmlieński's characterization of $\\perp_B^\\varepsilon$ is established. The forward and converse implications are governed by the distinct thresholds $\\frac{\\varepsilon}{(1-δ)^2}$ and $\\frac{\\varepsilon}{1-δ}$, which collapse to $\\varepsilon$ of Chmieliński precisely when $δ=0$, and the strictness of this gap is confirmed by counterexamples in $\\ell_\\infty^2$. A dual formulation of norming approximate orthogonality is established with a complete equivalence in the reflexive case. We apply our results to (vector-valued) continuous function spaces, which extends some earlier results and recovers few operator theoretical results with alternative proofs using measure theoretic techniques.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.18727",
  "author_total": 1,
  "authors": [
    {
      "name": "Saikat Roy"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1gah9fnyk72ax.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.18727",
  "fulltext_url": null,
  "id": "1gah9fnyk72ax",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.18727",
  "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": "W7165114004",
  "pdf_url": "https://arxiv.org/pdf/2606.18727",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1gah9fnyk72ax.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2606.18727",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.18727"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Norming Approximate Orthogonality in Normed Linear Spaces",
  "title_text": "Norming Approximate Orthogonality in Normed Linear Spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/1gah9fnyk72ax",
  "venue": "arXiv",
  "versions": []
}