发表机构
Claremont Graduate University（克莱蒙特研究生大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明子空间逼近格式中扩展局部自反性性质（ELRP）是多余的，通过二分法建立逼近数对偶性对所有有界线性算子无条件成立。
AI 中文摘要
我们通过研究子空间逼近格式如何与局部自反性相互作用，解决了算子逼近定量理论中的一个结构空白。近期文献在建立格式相对逼近数 $a_n(T,Q) = a_n(T^{**}, Q^{\perp\perp})$ 的对偶性时，依赖于将扩展局部自反性性质（ELRP）作为独立公理施加。在本文中，我们建立了一个二分法。对于无限维容许格式，我们证明它们自然生成完备的巢，完美满足 Oja-Veidenberg 局部自反性巢原理的拓扑假设。相反，我们证明对于具有有限维分量的逼近格式，双对偶几何会坍缩。因此，ELRP 完全是多余的，并且逼近数的对偶性对所有有界线性算子通过一个初等等距限制无条件成立。
英文摘要
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, $a_n(T,Q) = a_n(T^{**}, Q^{\perp\perp})$, 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.
