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Preface
Prologue
Contents of AMS Volume 109
1 Banach Spaces and Fixed-Point Theorems
1.1 Linear Spaces and Dimension
1.2 Normed Spaces and Convergence
1.3 Banach Spaces and the Cauchy Convergence Criterion
1.4 Open and Closed Sets
1.5 Operators
1.6 The Banach Fixed-Point Theorem and the Iteration Method
1.7 Applications to Integral Equations
1.8 Applications to Ordinary Differential Equations
1.9 Continuity
1.10 Convexity
1.11 Compactness
1.12 Finite-Dimensional Banach Spaces and Equivalent Norms
1.13 The Minkowski Functional and Homeomorphisms
1.14 The Brouwer Fixed-Point Theorem
1.15 The Schauder Fixed-Point Theorem
1.16 Applications to Integral Equations
1.17 Applications to Ordinary Differential Equations
1.18 The Leray-Schauder Principle and a priori Estimates
1.19 Sub-and Supersolutions, and the Iteration Method in Ordered Banach Spaces
1.20 Linear Operators
1.21 The Dual Space
1.22 Infinite Series in Normed Spaces
1.23 Banach Algebras and Operator Functions
1.24 Applications to Linear Differential Equations in Banach Spaces
1.25 Applications to the Spectrum
1.26 Density and Approximation
1.27 Summary of Important Notions
2 Hilbert Spaces, Orthogonality, and the Dirichlet
Principle
2.1 Hilbert Spaces
2.2 Standard Examples
2.3 Bilinear Forms
2.4 The Main Theorem on Quadratic Variational Problems
2.5 The Functional Analytic Justification of the Dirichlet Principle
2.6 The Convergence of the Ritz Method for Quadratic Variational Problems
2.7 Applications to Boundary-Value Problems, the Method of Finite Elements, and Elasticity
2.8 Generalized Functions and Linear Functionals
2.9 Orthogonal Projection
2.10 Linear Functionals and the Riesz Theorem
2.11 The Duality Map
2.12 Duality for Quadratic Variational Problems
2.13 The Linear Orthogonality Principle
2.14 Nonlinear Monotone Operators
2.15 Applications to the Nonlinear Lax-Milgram Theorem and the Nonlinear Orthogonality Principle
3 Hilbert Spaces and Generalized Fourier Series
3.1 Orthonormal Series
3.2 Applications to Classical Fourier Series
3.3 The Schmidt Orthogonalization Method
3.4 Applications to Polynomials
3.5 Unitary Operators
3.6 The Extension Principle
3.7 Applications to the Fourier Transformation
3.8 The Fourier Transform of Tempered Generalized Functions
4 Eigenvalue Problems for Linear Compact Symmetric Operators
¡­¡­
5 Self-Adjoint Operators, the Friedrichs Extension and the Partial Differential Equations of Mathematical physics
Epilogue
Appendix
References
Hints for Further Reading
List of Symbols
List of Theorems
List of the Most Important Definitions
Subject Index
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