Anisotropic Mixed-Norm Banach Spaces : Functional Analytic Structures, Operator Theory, Sobolev-Type Domains, Duality, And Isomorphism Of The Discrete Bravais Tori

This manuscript presents a comprehensive and rigorous study of Anisotropic Mixed-Norm Banach Spaces denoted Lᵖ(Rᵈ, qᵐ(C)),\,Lᵖ(Rᵈ, q²ᵐ(R)), establishing a unified framework that integrates Functional Analytic Structures, Operator Theory on the Position And Laplacian Operators X̂,\,Δ, Sobolev-Type Domains, Baire-Steinhaus Theorems, Topology, Dual Representation, and The Isomorphism Of Discrete Bravais Tori TᵈN₁⋯ Nd. This work establishes Theoretical Mathematical Foundations for the future study of Non-Hermitian Transport In Quantum Physics within this general class of Banach Spaces.