Applied Analysis

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Front Matter

     S Title

     APPLIED ANALYSIS

     Copyright (c) 2001 by World Scientific Publishing
          ISBN 9810241917
          QA300 .H93 2001 515--dc21
          LCCN 00051304

     Dedication

     Preface

     Contents

Chapter 1  Metric and Normed Spaces
     1.1 Metrics and norms
     1.2 Convergence
     1.3 Upper and lower bounds
     1.4 Continuity
     1.5 open and closed sets
     1.6 The completion of a metric space
     1.7 Compactness
     1.8 Maxima and minima
     1.9 References
     1.10 Exercises

Chapter 2  Continuous Functions
     2.1 Convergence of functions
     2.2 Spaces of continuous functions
     2.4 Compact subsets of C(K)
     2.5 Ordinary differential equations
     2.6 References
     2.7 Exercises

Chapter 3  The Contraction Mapping Theorem
     3.1 Contractions
     3.2 Fixed points of dynamical systems
     3.3 Integral equations
     3.4 Boundary value problems for differential equations
     3.5 Initial value problems for differential equations
     3.6 References
     3.7 Exercises

Chapter 4  Topological Spaces
     4.1 Topological spaces
     4.2 Bases of open sets
     4.3 Comparing topologies
     4.4 References
     4.5 Exercises

Chapter 5  Banach Spaces
     5.1 Banach spaces
     5.2 Bounded linear maps
     5.3 The kernel and range of a linear map
     5.4 Finite-dimensional Banach spaces
     5.5 Convergence of bounded operators
     5.6 Dual spaces
     5.7 References
     5.8 Exercises

Chapter 6  Hilbert Spaces
     6.1 Inner products
     6.2 orthogonality
     6.3 Orthonormal bases
     6.4 Hilbert spaces in applications
     6.5 References
     6.6 Exercises

Chapter 7  Fourier Series
     7.1 The Fourier basis
     7.2 Fourier series of differentiable functions
     7.3 The heat equation
     7.4 Other partial differential equations
     7.5 More applications of Fourier series
     7.6 Wavelets
     7.7 References
     7.8 Exercises

Chapter 8  Bounded Linear Operators on a Hilbert Space
     8.1 Orthogonal projections
     8.2 The dual of a Hilbert space
     8.3 The adjoint of an operator
     8.4 Self-adjoint and unitary operators
     8.5 The mean ergodic theorem
     8.6 Weak convergence in a Hilbert space
     8.7 References
     8.8 Exercises

Chapter 9  The Spectrum of Bounded Linear Operators
     9.1 Diagonalization of matrices
     9.2 The spectrum
     9.3 The spectral theorem for compact, self-adjoint operators
     9.4 Compact operators
     9.5 Functions of operators
     9.6 Perturbation of eigenvalues
     9.7 References
     9.8 Exercises

Chapter 10  Linear Differential Operators and Green\'s Functions
     10.1 Unbounded operators
     10.2 The adjoint of a differential operator
     10.3 Green\'s functions
     10.4 Weak derivatives
     10.5 The Sturm-Liouville eigenvalue problem
     10.6 Laplace\'s equation
     10.7 References
     10.8 Exercises

Chapter 11  Distributions and the Fourier Transform
     11.1 The Schwartz space
     11.2 Tempered distributions
     11.3 Operations on distributions
     11.4 The convergence of distributions
     11.5 The Fourier transform of test functions
     11.6 The Fourier transform of tempered distributions
     11.8 The Fourier transform on L2
     11.9 Translation invariant operators
     11.10 Green\'s functions
     11.11 The Poisson summation formula
     11.12 The central limit theorem
     11.13 References
     11.14 Exercises

Chapter 12  Measure Theory and Function Spaces
     12.1 Measures
     12.2 Measurable functions
     12.3 Integration
     12.4 Convergence theorems
     12.5 Product measures and Fubini\'s theorem
     12.6 The Lp spaces
     12.7 The basic inequalities
     12.8 The dual space of Lp
     12.9 Sobolev spaces
     12.10 Properties of Sobolev spaces
     12.11 Laplace\'s equation
     12.12 References
     12.13 Exercises

Chapter 13  Differential Calculus and Variational Methods
     13.1 Linearization
     13.2 Vector-valued integrals
     13.3 Derivatives of maps on Banach spaces
     13.4 The inverse and implicit function theorems
     13.5 Newton\'s method
     13.6 Linearized stability
     13.7 The calculus of variations
     13.8 Hamilton\'s equation and classical mechanics
     13.9 Multiple integrals in the calculus of variations
     13.10 References
     13.11 Exercises

Back Matter
     Bibliography
     Index

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