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 Arzel