Applied Functional Analysis

Preface Part I. Distribution Theory and Green's Functions Chapter 1. Generalised Functions 1.1 The Delta function 1.2 Basic distribution theory 1.3 Operations on distributions 1.4 Convergence of distributions 1.5 Further developments 1.6 Fourier Series and the Poisson Sum formula 1.7 Summary and References Problems Chapter 2. Differential Equations and Green's Functions 2.1 The Integral of a distribution 2.2 Linear differential equations 2.3 Fundamental solutions of differential equations 2.4 Green's functions 2.5 Applications of Green's functions 2.6 Summary and References Problems Chapter 3. Fourier Transforms and Partial differential Equations 3.1 The classical Fourier transform 3.2 Distributions of slow growth 3.3 Generalised Fourier transforms 3.4 Generalised functions of several variables 3.5 Green's function for the Laplacian 3.6 Green's function for the Three-dimensional wave equation 3.7 Summary and References Problems Part II. Banach spaces and fixed point theorems Chapter 4. Normed spaces 4.1 Vector spaces 4.2 Normed spaces 4.3 Convergence 4.4 Open and closed sets 4.5 Completeness 4.6 Equivalent norms 4.7 Summary and References Problems Chapter 5. The contraction mapping theorem 5.1 Operators on Vector spaces 5.2 The contraction mapping theorem 5.3 Application to differential and integral equations 5.4 Nonlinear diffusive equilibrium 5.5 Nonlinear diffusive equilibrium in three dimensions 5.6 Summary and References Problems Chapter 6. Compactness and Schauder's theorem 6.1 Continuous operators 6.2 Brouwer's theorem 6.3 Compactness 6.4 Relative compactness 6.5 Arzelà's theorem 6.6 Schauder's theorems 6.7 Forced nonlinear oscillations 6.8 Swirling flow 6.9 Summary and References Problems Part III. Operators in Hilbert Space Chapter 7. Hilbert space 7.1 Inner product spaces 7.2 Orthogonal bases 7.3 Orthogonal expansions 7.4 The Bessel, Parseval, and Riesz-Fischer theorems 7.5 Orthogonal decomposition 7.6 Functionals on normed spaces 7.7 Functionals in Hilbert space 7.8 Weak convergence 7.9 Summary and References Problems Chapter 8. The Theory of operators 8.1 Bounded operators on normed spaces 8.2 The algebra of bounded operators 8.3 Self-adjoint operators 8.4 Eigenvalue problems for self-adjoint operators 8.5 Compact operators 8.6 Summary and References Problems Chapter 9. The Spectral theorem 9.1 The spectral theorem 9.2 Sturm-Liouville systems 9.3 Partial differential equations 9.4 The Fredholm alternative 9.5 Projection operators 9.6 Summary and References Problems Chapter 10. Variational methods 10.1 Positive operators 10.2 Approximation to the first eigenvalue 10.3 The Rayleigh-Ritz method for eigenvalues 10.4 The theory of the Rayleigh-Ritz method 10.5 Inhomogeneous Equations 10.6 Complementary bounds 10.7 Summary and References Problems Part IV. Further developments Chapter 11. The differential calculus of operators and its applications 11.1 The Fréchet derivative 11.2 Higher derivatives 11.3 Maxima and Minima 11.4 Linear stability theory 11.5. Nonlinear stability 11.6 Bifurcation theory 11.7 Bifurcation and stability 11.8 Summary and References Chapter 12. Distributional Hilbert spaces 12.1 The space of square-integrable distributions 12.2 Sobolev spaces 12.3 Application to partial differential equations 12.4 Summary and References Appendix A. Sets and mappings Appendix B. Sequences, series, and uniform convergence Appendix C. Sup and inf Appendix D. Countability Appendix E. Equivalence relations Appendix F. Completion Appendix G. Sturm-Liouville systems Appendix H. Fourier's theorem Appendix I. Proofs of 9.24 and 9.25 Notes on the Problems; Supplementary Problems; Symbol index; References and name index; Subject index