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