Functional Analysis and Applied Optimization in Banach Spaces : Applications to Non Convex Variational Models This book introduces the basic concepts of real and functional analysis. It presents the fundamentals of the calculus of variations convex analysis duality and optimization that are necessary to develop applications to physics and engineering problems. The book includes introductory and advanced concepts in measure and integration as well as an introduction to Sobolev spaces. The problems presented are nonlinear with non convex variational formulation. Notably the primal global minima may not be attained in some situations in which cases the solution of the dual problem corresponds to an appropriate weak cluster point of minimizing sequences for the primal one. Indeed the dual approach more readily facilitates numerical computations for some of the selected models. While intended primarily for applied mathematicians the text will also be of interest to engineers physicists and other researchers in related fields 1. Topological Vector Spaces 2. The Hahn Bananch Theorems and Weak Topologies 3. Topics on Linear Operators 4. Basic Results on Measure and Integration. 5. The Lebesgue Measure in Rn 6. Other Topics in Measure and Integration 7. Distributions 8. The Lebesque and Sobolev Spaces. 9. Basic Concepts on the Calculus of Variations 10. Basic Concepts on Convex Analysis 11. Constrained Variational Analysis 12. Duality Applied to Elasticity 13. Duality Applied to a Plate Model 14. About Ginzburg Landau Type Equations: The Simpler Real Case. 15. Full Complex Ginzburg Landau System. 16. More on Duality and Computation in the Ginzburg Landau System. 17. On Duality Principles for Scalar and Vectorial Multi Well Variational Problems 18. More on Duality Principles for Multi Well Problems 19. Duality and Computation for Quantum Mechanics Models 20. Duality Applied to the Optimal Design in Elasticity 21. Duality Applied to Micro magnetism 22. The Generalized Method of Lines Applied to Fluid Mechanics 23. Duality Applied to the Optimal Control and Optimal Design of a Beam Model