Computational Theory of Norm-Attaining Functionals: Algorithms, Stability, and Applications in Banach Spaces

Abstract

This paper develops novel computational methods for studying norm-attaining functionals in infinite-dimensional Banach spaces. We present constructive approximation algorithms with explicit convergence rates, stability analysis under discretization and perturbations, and new geometric characterizations of norm attainment. Key results include: (1) efficient procedures to compute norm-attaining approximations of functionals in uniformly convex spaces, with quantitative error bounds; (2) stability theorems for finite-dimensional projections in reflexive spaces; (3) perturbation resilience estimates relating to the modulus of convexity; and (4) applications to PDE-constrained optimization and functional regression. Our approach combines techniques from functional analysis, approximation theory, and computational mathematics, yielding both theoretical insights and practical algorithms. The results significantly extend the classical Bishop-Phelps theorem by providing computable versions and quantitative estimates in various Banach space geometries.