| PrefacePrologueContents of AMS Volume 1091 Banach Spaces and Fixed-Point Theorems1.1 Linear Spaces and Dimension1.2 Normed Spaces and Convergence1.3 Banach Spaces and the Cauchy Convergence Criterion1.4 Open and Closed Sets1.5 Operators1.6 The Banach Fixed-Point Theorem and the Iteration Method1.7 Applications to Integral Equations1.8 Applications to Ordinary Differential Equations1.9 Continuity1.10 Convexity1.11 Compactness1.12 Finite-Dimensional Banach Spaces and Equivalent Norms1.13 The Minkowski Functional and Homeomorphisms1.14 The Brouwer Fixed-Point Theorem1.15 The Schauder Fixed-Point Theorem1.16 Applications to Integral Equations1.17 Applications to Ordinary Differential Equations1.18 The Leray-Schauder Principle and a priori Estimates1.19 Sub-and Supersolutions, and the Iteration Method in Ordered Banach Spaces1.20 Linear Operators1.21 The Dual Space1.22 Infinite Series in Normed Spaces1.23 Banach Algebras and Operator Functions1.24 Applications to Linear Differential Equations in Banach Spaces1.25 Applications to the Spectrum1.26 Density and Approximation1.27 Summary of Important Notions2 Hilbert Spaces, Orthogonality, and the DirichletPrinciple2.1 Hilbert Spaces2.2 Standard Examples2.3 Bilinear Forms2.4 The Main Theorem on Quadratic Variational Problems2.5 The Functional Analytic Justification of the Dirichlet Principle2.6 The Convergence of the Ritz Method for Quadratic Variational Problems2.7 Applications to Boundary-Value Problems, the Method of Finite Elements, and Elasticity2.8 Generalized Functions and Linear Functionals2.9 Orthogonal Projection2.10 Linear Functionals and the Riesz Theorem2.11 The Duality Map2.12 Duality for Quadratic Variational Problems2.13 The Linear Orthogonality Principle2.14 Nonlinear Monotone Operators2.15 Applications to the Nonlinear Lax-Milgram Theorem and the Nonlinear Orthogonality Principle3 Hilbert Spaces and Generalized Fourier Series3.1 Orthonormal Series3.2 Applications to Classical Fourier Series3.3 The Schmidt Orthogonalization Method3.4 Applications to Polynomials3.5 Unitary Operators3.6 The Extension Principle3.7 Applications to the Fourier Transformation3.8 The Fourier Transform of Tempered Generalized Functions4 Eigenvalue Problems for Linear Compact Symmetric Operators4.1 Symmetric Operators4.2 The Hilbert-Schmidt Theory4.3 The Fredholm Alternative4.4 Applications to Integral Equations4.5 Applications to Boundary-Eigenvalue Value Problems5 Self-Adjoint Operators, the Priedrichs Extension and the Partial Differential Equations of MathematicalPhysics5.1 Extensions and Embeddings5.2 Self-Adjoint Operators5.3 The Energetic Space5.4 The Energetic Extension5.5 The Friedrichs Extension of Symmetric Operators5.6 Applications to Boundary-Eigenvalue Problems for the Laplace Equation5.7 The Poincar6 Inequality and Rellich's Compactness Theorem5.8 Functions of Self-Adjoint Operators5.9 Semigroups, One-Parameter Groups, and Their Physical Relevance5.10 Applications to the Heat Equation5.11 Applications to the Wave Equation5.12 Applications to the Vibrating String and the Fourier Method5.13 Applications to the SchrSdinger Equation5.14 Applications to Quantum Mechanics5.15 Generalized Eigenfunctions5.16 Trace Class Operators5.17 Applications to Quantum Statistics5.18 C*-Algebras and the Algebraic Approach to Quantum Statistics5.19 The Fock Space in Quantum Field Theory and the Pauli Principle5.20 A Look at Scattering Theory5.21 The Language of Physicists in Quantum Physics and the Justification of the Dirac Calculus5.22 The Euclidean Strategy in Quantum Physics5.23 Applications to Feynman's Path Integral5.24 The Importance of the Propagator in Quantum Physics 5.25 A Look at Solitons and Inverse Scattering TheoryEpilogueAppendixReferencesHints for Further ReadingList of SymbolsList of TheoremsList of the Most Important DefinitionsSubject Index |
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