Local Reflexivity and Duality for Subspace Approximation Schemes
Abstract
We address a structural gap in the quantitative theory of operator approximation by investigating how subspace approximation schemes interact with local reflexivity. Recent literature establishing the duality of scheme-relative approximation numbers, an(T,Q) = an(T**, Q⊥⊥), has relied on imposing an Extended Local Reflexivity Property (ELRP) as an independent axiom. In this note, we establish a dichotomy. For infinite-dimensional admissible schemes, we prove they naturally generate complete nests, perfectly satisfying the topological hypotheses of the Oja-Veidenberg Nest Principle of Local Reflexivity. Conversely, we demonstrate that for approximation schemes with finite-dimensional components, the bidual geometry collapses. Consequently, the ELRP is entirely superfluous, and the duality of approximation numbers holds unconditionally for all bounded linear operators via an elementary isometric restriction.
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BibTeX
@misc{thiong2026local,
title = {{Local Reflexivity and Duality for Subspace Approximation Schemes}},
author = {Daniel Akech Thiong},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10460},
eprint = {2609.10460},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10460},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Local Reflexivity and Duality for Subspace Approximation Schemes
AU - Daniel Akech Thiong
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10460
UR - https://arxiv.org/abs/2609.10460
AB - We address a structural gap in the quantitative theory of operator approximation by investigating how subspace approximation schemes interact with local reflexivity. Recent literature establishing the duality of scheme-relative approximation numbers, a_n(T,Q) = a_n(T^**, Q^⊥⊥), has relied on imposing an Extended Local Reflexivity Property (ELRP) as an independent axiom. In this note, we establish a dichotomy. For infinite-dimensional admissible schemes, we prove they naturally generate complete nests, perfectly satisfying the topological hypotheses of the Oja-Veidenberg Nest Principle of Local Reflexivity. Conversely, we demonstrate that for approximation schemes with finite-dimensional components, the bidual geometry collapses. Consequently, the ELRP is entirely superfluous, and the duality of approximation numbers holds unconditionally for all bounded linear operators via an elementary isometric restriction.
ER -