Author | Zeidler, Eberhard. author |
---|---|
Title | Applied Functional Analysis [electronic resource] : Main Principles and Their Applications / by Eberhard Zeidler |
Imprint | New York, NY : Springer New York : Imprint: Springer, 1995 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-0821-1 |
Descript | XVI, 406 p. online resource |
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[a,b]* -- 1.4 Applications to the Moment Problem -- 1.5 Minimum Norm Problems and Duality Theory -- 1.6 Applications to ?ebyลกev 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 Frรฉchet 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