Approximate Bases and Interpolation in Non-Separable Banach Spaces: A Unified Analytical Framework

Abstract

Juan Alberto Molina Garcia*

This paper develops a unified analytical framework for non-separable Banach spaces (NSBS) grounded in the concepts of approximate Schauder bases, weakly compact approximation operators, and local–global interpolation structures. The traditional limitations associated with the absence of separability and sequential compactness are overcome by replacing sequences with directed nets and by localising all analytical arguments within separable hulls. Within this setting, the notions of approximate interpolation couples and approximate real and complex interpolation spaces are introduced and analysed in detail. We establish that boundedness, compactness, duality, and stability properties extend naturally from the classical separable case, providing new generalisations of the Lions–Peetre and Riesz–Thorin theorems. Spectral theory is reformulated for bounded linear operators in NSBS, proving spectral stability under approximate compactness and continuity of spectra under analytic perturbations. Applications to operator theory and mathematical physics are discussed, including spectral–interpolation correspondences, weak compactness criteria, and models for quantum and statistical systems where non-separable structures arise intrinsically. These results yield a coherent generalisation of fundamental principles of functional analysis and open new avenues for research in operator algebras, evolution equations, and non-separable

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