This book provides an introduction to those parts of analysis that are most useful in applications for graduate students. The material is selected for use in applied problems, and is presented clearly and simply but without sacrificing mathematical rigor.
The text is accessible to students from a wide variety of backgrounds, including undergraduate students entering applied mathematics from non-mathematical fields and graduate students in the sciences and engineering who want to learn analysis. A basic background in calculus, linear algebra and ordinary differential equations, as well as some familiarity with functions and sets, should be sufficient.
Readership: Graduate students in applied analysis.
Author(s): John K Hunter
Publisher: World Scientific
Year: 2001
Language: English
Pages: C, xii+439, B
Cover
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
Back Cover
