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, $a_n(T,Q) = a_n(T^{**}, Q^{\perp\perp})$, 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.