Norming Approximate Orthogonality in Normed Linear Spaces
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, ε ∈ [0,1) with ε < (1−δ)2, we say x nperp y in X if there exists f ∈ X* with |f(x)| ≥ (1−δ)|f||x| and |f(y)| ≤ ε/(1−δ)|f||y|, simultaneously relaxing both the norming condition on x and the vanishing condition on y. It is proved that xnperp y ⟺ |x+λy|≥ (1−δ)|x|−ε/(1−δ)|λy| ∀ 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= ε = 0, the approximate orthogonality of Chmieliński at δ= 0, and the approximate orthogonality of Dragomir at ε = 0. A two-parameter proximity result generalizing Chmlieński's characterization of ⊥Bε is established. The forward and converse implications are governed by the distinct thresholds ε/((1−δ)2) and ε/(1−δ), which collapse to ε of Chmieliński precisely when δ=0, and the strictness of this gap is confirmed by counterexamples in ℓ∞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.
Related
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BibTeX
@misc{roy2026norming,
title = {{Norming Approximate Orthogonality in Normed Linear Spaces}},
author = {Saikat Roy},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.18727},
eprint = {2606.18727},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.18727},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Norming Approximate Orthogonality in Normed Linear Spaces
AU - Saikat Roy
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2606.18727
UR - https://arxiv.org/abs/2606.18727
AB - 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, ε ∈ [0,1) with ε < (1−δ)^2, we say x nperp y in X if there exists f ∈ X^* with |f(x)| ≥ (1−δ)|f||x| and |f(y)| ≤ ε/(1−δ)|f||y|, simultaneously relaxing both the norming condition on x and the vanishing condition on y. It is proved that xnperp y ⟺ |x+λy|≥ (1−δ)|x|−ε/(1−δ)|λy| ∀ 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= ε = 0, the approximate orthogonality of Chmieliński at δ= 0, and the approximate orthogonality of Dragomir at ε = 0. A two-parameter proximity result generalizing Chmlieński's characterization of ⊥_B^ε is established. The forward and converse implications are governed by the distinct thresholds ε/((1−δ)^2) and ε/(1−δ), which collapse to ε of Chmieliński precisely when δ=0, and the strictness of this gap is confirmed by counterexamples in ℓ_∞^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.
ER -