1 The Hahn-Banach Theorem Optimization Problems. - 1. 1 The Hahn-Banach Theorem. - 1. 2 Applications to the Separation of Convex Sets. - 1. 3 The Dual Space C[ab]*. - 1. 4 Applications to the Moment Problem. - 1. 5 Minimum Norm Problems and Duality Theory. - 1. 6 Applications to ?ebyšev Approximation. - 1. 7 Applications to the Optimal Control of Rockets. - 2 Variational Principles and Weak Convergence. - 2. 1 The nth Variation. - 2. 2 Necessary and Sufficient Conditions for Local Extrema and the Classical Calculus of Variations. - 2. 3 The Lack of Compactness in Infinite-Dimensional Banach Spaces. - 2. 4 Weak Convergence. - 2. 5 The Generalized Weierstrass Existence Theorem. - 2. 6 Applications to the Calculus of Variations. - 2. 7 Applications to Nonlinear Eigenvalue Problems. - 2. 8 Reflexive Banach Spaces. - 2. 9 Applications to Convex Minimum Problems and Variational Inequalities. - 2. 10 Applications to Obstacle Problems in Elasticity. - 2. 11 Saddle Points. - 2. 12 Applications to Duality Theory. - 2. 13 The von Neumann Minimax Theorem on the Existence of Saddle Points. - 2. 14 Applications to Game Theory. - 2. 15 The Ekeland Principle about Quasi-Minimal Points. - 2. 16 Applications to a General Minimum Principle via the Palais-Smale Condition. - 2. 17 Applications to the Mountain Pass Theorem. - 2. 18 The Galerkin Method and Nonlinear Monotone Operators. - 2. 19 Symmetries and Conservation Laws (The Noether Theorem). - 2. 20 The Basic Ideas of Gauge Field Theory. - 2. 21 Representations of Lie Algebras. - 2. 22 Applications to Elementary Particles. - 3 Principles of Linear Functional Analysis. - 3. 1 The Baire Theorem. - 3. 2 Application to the Existence of Nondifferentiable Continuous Functions. - 3. 3 The Uniform Boundedness Theorem. - 3. 4 Applications to Cubature Formulas. - 3. 5 The Open Mapping Theorem. - 3. 6 Product Spaces. - 3. 7 The Closed Graph Theorem. - 3. 8 Applications to Factor Spaces. - 3. 9 Applications to Direct Sums and Projections. - 3. 10 Dual Operators. - 3. 11 The Exactness of the Duality Functor. - 3. 12 Applications to the Closed Range Theorem and to Fredholm Alternatives. - 4 The Implicit Function Theorem. - 4. 1 m-Linear Bounded Operators. - 4. 2 The Differential of Operators and the Fréchet Derivative. - 4. 3 Applications to Analytic Operators. - 4. 4 Integration. - 4. 5 Applications to the Taylor Theorem. - 4. 6 Iterated Derivatives. - 4. 7 The Chain Rule. - 4. 8 The Implicit Function Theorem. - 4. 9 Applications to Differential Equations. - 4. 10 Diffeomorphisms and the Local Inverse Mapping Theorem. - 4. 11 Equivalent Maps and the Linearization Principle. - 4. 12 The Local Normal Form for Nonlinear Double Splitting Maps. - 4. 13 The Surjective Implicit Function Theorem. - 4. 14 Applications to the Lagrange Multiplier Rule. - 5 Fredholm Operators. - 5. 1 Duality for Linear Compact Operators. - 5. 2 The Riesz-Schauder Theory on Hilbert Spaces. - 5. 3 Applications to Integral Equations. - 5. 4 Linear Fredholm Operators. - 5. 5 The Riesz-Schauder Theory on Banach Spaces. - 5. 6 Applications to the Spectrum of Linear Compact Operators. - 5. 7 The Parametrix. - 5. 8 Applications to the Perturbation of Fredholm Operators. - 5. 9 Applications to the Product Index Theorem. - 5. 10 Fredholm Alternatives via Dual Pairs. - 5. 11 Applications to Integral Equations and Boundary-Value Problems. - 5. 12 Bifurcation Theory. - 5. 13 Applications to Nonlinear Integral Equations. - 5. 14 Applications to Nonlinear Boundary-Value Problems. - 5. 15 Nonlinear Fredholm Operators. - 5. 16 Interpolation Inequalities. - 5. 17 Applications to the Navier-Stokes Equations. - References. - List of Symbols. - List of Theorems. - List of Most Important Definitions. Language: English