On Linear Metric Projections in Non-Reflexive Banach Spaces: Proximal Lattices, Multi-Valued Boundaries, and Linear Selections (Part 4)

This paper establishes a unified geometric framework for the construction of continuous linear selections operating over set-valued metric projections within non-reflexive Banach spaces, specifically L1(Ω) and C(K). Extending the structural limits analyzed in previous parts of this series, we resolve the mathematical barriers induced by flat unit sphere facets within non-reflexive topologies. We restrict our architectural scope to proximinal subspace lattices and derive the exact analytical conditions for the existence of unique linear selections using Birkhoff–James orthogonality structures and Chebyshev alternation subsets. Finally, we prove a direct-sum topological decomposition theorem and establish Hausdorff semi-continuity bounds, providing complete regularized closure for non-smooth high-dimensional computational optimization frameworks.