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| Preface | |
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| Distribution Theory and Green's Functions | |
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| Generalised Functions | |
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| The Delta Function | |
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| Basic Distribution Theory | |
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| Operations on Distributions | |
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| Convergence of Distributions | |
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| Further Developments | |
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| Fourier Series and the Poisson Sum Formula | |
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| Summary and References | |
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| Problems | |
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| Differential Equations and Green's Functions | |
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| The Integral of a Distribution | |
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| Linear Differential Equations | |
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| Fundamental Solutions of Differential Equations | |
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| Green's Functions | |
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| Applications of Green's Functions | |
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| Summary and References | |
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| Problems | |
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| Fourier Transforms and Partial Differential Equations | |
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| The Classical Fourier Transform | |
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| Distributions of Slow Growth | |
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| Generalised Fourier Transforms | |
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| Generalised Functions of Several Variables | |
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| Green's Function for the Laplacian | |
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| Green's Function for the Three-Dimensional Wave Equation | |
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| Summary and References | |
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| Problems | |
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| Banach Spaces and Fixed Point Theorems | |
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| Normed Spaces | |
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| Vector Spaces | |
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| Normed Spaces | |
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| Convergence | |
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| Open and Closed Sets | |
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| Completeness | |
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| Equivalent Norms | |
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| Summary and References | |
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| Problems | |
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| The Contraction Mapping Theorem | |
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| Operators on Vector Spaces | |
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| The Contraction Mapping Theorem | |
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| Application to Differential and Integral Equations | |
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| Nonlinear Diffusive Equilibrium | |
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| Nonlinear Diffusive Equilibrium in Three Dimensions | |
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| Summary and References | |
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| Problems | |
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| Compactness and Schauder's Theorem | |
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| Continuous Operators | |
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| Brouwer's Theorem | |
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| Compactness | |
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| Relative Compactness | |
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| Arzela's Theorem | |
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| Schauder's Theorems | |
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| Forced Nonlinear Oscillations | |
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| Swirling Flow | |
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| Summary and References | |
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| Problems | |
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| Operators in Hilbert Space | |
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| Hilbert Space | |
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| Inner Product Spaces | |
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| Orthogonal Bases | |
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| Orthogonal Expansions | |
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| The Bessel, Parseval, and Riesz-Fischer Theorems | |
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| Orthogonal Decomposition | |
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| Functionals on Normed Spaces | |
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| Functionals in Hilbert Space | |
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| Weak Convergence | |
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| Summary and References | |
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| Problems | |
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| The Theory of Operators | |
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| Bounded Operators on Normed Spaces | |
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| The Algebra of Bounded Operators | |
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| Self-Adjoint Operators | |
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| Eigenvalue Problems for Self-Adjoint Operators | |
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| Compact Operators | |
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| Summary and References | |
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| Problems | |
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| The Spectral Theorem | |
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| The Spectral Theorem | |
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| Sturm-Liouville Systems | |
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| Partial Differential Equations | |
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| The Fredholm Alternative | |
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| Projection Operators | |
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| Summary and References | |
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| Problems | |
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| Variational Methods | |
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| Positive Operators | |
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| Approximation to the First Eigenvalue | |
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| The Rayleigh-Ritz Method for Eigenvalues | |
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| The Theory of the Rayleigh-Ritz Method | |
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| Inhomogeneous Equations | |
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| Complementary Bounds | |
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| Summary and References | |
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| Problems | |
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| Further Developments | |
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| The Differential Calculus of Operators and its Applications | |
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| The Frechet Derivative | |
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| Higher Derivatives | |
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| Maxima and Minima | |
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| Linear Stability Theory | |
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| Nonlinear Stability | |
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| Bifurcation Theory | |
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| Bifurcation and Stability | |
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| Summary and References | |
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| Distributional Hilbert Spaces | |
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| The Space of Square-Integrable Distributions | |
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| Sobolev Spaces | |
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| Application to Partial Differential Equations | |
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| Summary and References | |
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| Appendices | |
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| Sets and Mappings | |
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| Sequences, Series, and Uniform Convergence | |
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| Sup and Inf | |
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| Countability | |
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| Equivalence Relations | |
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| Completion | |
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| Sturm-Liouville Systems | |
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| Fourier's Theorem | |
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| Proofs of 9.24 and 9.25 | |
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| Notes on the Problems | |
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| Supplementary Problems | |
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| Symbol Index | |
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| References and Name Index | |
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| Subject Index | |