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 \textbf{Anisotropic Mixed-Norm Banach Spaces} denoted $\mathbf{L}^p(\mathbb{R}^d, \ell_q^m(\mathbb{C})),\,\mathbf{L}^p(\mathbb{R}^d, \ell_q^{2m}(\mathbb{R}))$, establishing a unified framework that integrates \textbf{Functional Analytic Structures}, \textbf{Operator Theory} on the \textbf{Position And Laplacian Operators} $\boldsymbol{\mathfrak{\hat{X}}},\,\boldsymbol{\hat\Delta}$, \textbf{Sobolev-Type Domains}, \textbf{Baire-Steinhaus Theorems}, \textbf{Topology}, \textbf{Dual Representation}, and The \textbf{Isomorphism Of Discrete Bravais Tori} $\mathbb{T}^d_{N_1\cdots N_d}$. This work establishes \textbf{Theoretical Mathematical Foundations} for the future study of \textbf{Non-Hermitian Transport In Quantum Physics} within this general class of \textbf{Banach Spaces}.