| Preface | p. 9 |
| Distribution Theory and Green's Functions | |
| Generalised Functions | |
| The Delta Function | p. 12 |
| Basic Distribution Theory | p. 15 |
| Operations on Distributions | p. 19 |
| Convergence of Distributions | p. 24 |
| Further Developments | p. 26 |
| Fourier Series and the Poisson Sum Formula | p. 30 |
| Summary and References | p. 33 |
| Problems | p. 33 |
| Differential Equations and Green's Functions | |
| The Integral of a Distribution | p. 37 |
| Linear Differential Equations | p. 39 |
| Fundamental Solutions of Differential Equations | p. 41 |
| Green's Functions | p. 45 |
| Applications of Green's Functions | p. 48 |
| Summary and References | p. 51 |
| Problems | p. 51 |
| Fourier Transforms and Partial Differential Equations | |
| The Classical Fourier Transform | p. 53 |
| Distributions of Slow Growth | p. 56 |
| Generalised Fourier Transforms | p. 60 |
| Generalised Functions of Several Variables | p. 64 |
| Green's Function for the Laplacian | p. 67 |
| Green's Function for the Three-Dimensional Wave Equation | p. 74 |
| Summary and References | p. 78 |
| Problems | p. 78 |
| Banach Spaces and Fixed Point Theorems | |
| Normed Spaces | |
| Vector Spaces | p. 84 |
| Normed Spaces | p. 91 |
| Convergence | p. 95 |
| Open and Closed Sets | p. 98 |
| Completeness | p. 104 |
| Equivalent Norms | p. 110 |
| Summary and References | p. 112 |
| Problems | p. 112 |
| The Contraction Mapping Theorem | |
| Operators on Vector Spaces | p. 116 |
| The Contraction Mapping Theorem | p. 120 |
| Application to Differential and Integral Equations | p. 123 |
| Nonlinear Diffusive Equilibrium | p. 128 |
| Nonlinear Diffusive Equilibrium in Three Dimensions | p. 131 |
| Summary and References | p. 134 |
| Problems | p. 134 |
| Compactness and Schauder's Theorem | |
| Continuous Operators | p. 139 |
| Brouwer's Theorem | p. 144 |
| Compactness | p. 149 |
| Relative Compactness | p. 152 |
| Arzela's Theorem | p. 155 |
| Schauder's Theorems | p. 158 |
| Forced Nonlinear Oscillations | p. 160 |
| Swirling Flow | p. 165 |
| Summary and References | p. 170 |
| Problems | p. 171 |
| Operators in Hilbert Space | |
| Hilbert Space | |
| Inner Product Spaces | p. 176 |
| Orthogonal Bases | p. 180 |
| Orthogonal Expansions | p. 183 |
| The Bessel, Parseval, and Riesz-Fischer Theorems | p. 188 |
| Orthogonal Decomposition | p. 192 |
| Functionals on Normed Spaces | p. 195 |
| Functionals in Hilbert Space | p. 198 |
| Weak Convergence | p. 199 |
| Summary and References | p. 205 |
| Problems | p. 206 |
| The Theory of Operators | |
| Bounded Operators on Normed Spaces | p. 210 |
| The Algebra of Bounded Operators | p. 214 |
| Self-Adjoint Operators | p. 221 |
| Eigenvalue Problems for Self-Adjoint Operators | p. 226 |
| Compact Operators | p. 230 |
| Summary and References | p. 234 |
| Problems | p. 235 |
| The Spectral Theorem | |
| The Spectral Theorem | p. 240 |
| Sturm-Liouville Systems | p. 245 |
| Partial Differential Equations | p. 252 |
| The Fredholm Alternative | p. 257 |
| Projection Operators | p. 260 |
| Summary and References | p. 268 |
| Problems | p. 269 |
| Variational Methods | |
| Positive Operators | p. 274 |
| Approximation to the First Eigenvalue | p. 277 |
| The Rayleigh-Ritz Method for Eigenvalues | p. 281 |
| The Theory of the Rayleigh-Ritz Method | p. 284 |
| Inhomogeneous Equations | p. 292 |
| Complementary Bounds | p. 296 |
| Summary and References | p. 301 |
| Problems | p. 301 |
| Further Developments | |
| The Differential Calculus of Operators and its Applications | |
| The Frechet Derivative | p. 308 |
| Higher Derivatives | p. 312 |
| Maxima and Minima | p. 316 |
| Linear Stability Theory | p. 319 |
| Nonlinear Stability | p. 323 |
| Bifurcation Theory | p. 326 |
| Bifurcation and Stability | p. 330 |
| Summary and References | p. 334 |
| Distributional Hilbert Spaces | |
| The Space of Square-Integrable Distributions | p. 336 |
| Sobolev Spaces | p. 340 |
| Application to Partial Differential Equations | p. 343 |
| Summary and References | p. 345 |
| Appendices | |
| Sets and Mappings | p. 347 |
| Sequences, Series, and Uniform Convergence | p. 348 |
| Sup and Inf | p. 352 |
| Countability | p. 354 |
| Equivalence Relations | p. 357 |
| Completion | p. 357 |
| Sturm-Liouville Systems | p. 360 |
| Fourier's Theorem | p. 362 |
| Proofs of 9.24 and 9.25 | p. 364 |
| Notes on the Problems | p. 367 |
| Supplementary Problems | p. 375 |
| Symbol Index | p. 379 |
| References and Name Index | p. 382 |
| Subject Index | p. 387 |
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