Davidson K.R., Donsig A.P. Real Analysis with Real Applications
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Real Analysis
with
Real Applications
Real Analysis
with
Real Applications
Kenneth R. Davidson
University of Waterloo
Allan P. Donsig
University of Nebraska
Prentice Hall
Upper Saddle River, NJ 07458
Library of Congress Cataloging–in–Publication Data
Davidson, Kenneth R.
Real analysis with real applications/Kenneth R. Davidson, Allan P. Donsig.
p. cm.
Includes bibliographical references and index.
ISBN 0-13-041647-9
1. Mathematical analysis. I. Donsig, Allan P. II. Title.
QA 300 .D342 2002
515–dc21
2001052318
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Hart House Permanent Collection, University of Toronto.
c
°2002 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
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ISBN 0-13-041647-9
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CONTENTS
Preface xi
Dependence on Earlier Sections xiv
Possible Course Outlines xvi
1 Background 1
1.1 The Language of Mathematics 1
1.2 Sets and Functions 5
1.3 Calculus 10
1.4 Linear Algebra 11
1.5 The Role of Proofs 19
•1.6 Appendix: Equivalence Relations 25
Part A Abstract Analysis 29
2 The Real Numbers 31
2.1 An Overview of the Real Numbers 31
2.2 Infinite Decimals 34
2.3 Limits 37
2.4 Basic Properties of Limits 42
2.5 Upper and Lower Bounds 46
2.6 Subsequences 51
2.7 Cauchy Sequences 55
•2.8 Appendix: Cardinality 60
3 Series 66
3.1 Convergent Series 66
3.2 Convergence Tests for Series 70
•3.3 The Number e 77
•3.4 Absolute and Conditional Convergence 80
4 The Topology of R
n
88
4.1 n-dimensional Space 88
4.2 Convergence and Completeness in R
n
92
4.3 Closed and Open Subsets of R
n
96
4.4 Compact Sets and the Heine–Borel Theorem 101
vii
viii Contents
5 Functions 108
5.1 Limits and Continuity 108
5.2 Discontinuous Functions 114
5.3 Properties of Continuous Functions 120
5.4 Compactness and Extreme Values 124
5.5 Uniform Continuity 127
5.6 The Intermediate Value Theorem 133
•5.7 Monotone Functions 135
6 Differentiation and Integration 141
6.1 Differentiable Functions 141
6.2 The Mean Value Theorem 148
6.3 Riemann Integration 153
6.4 The Fundamental Theorem of Calculus 164
•6.5 Wallis’s Product and Stirling’s Formula 169
? 6.6 Measure Zero and Lebesgue’s Theorem 175
7 Normed Vector Spaces 179
7.1 Definition and Examples 179
7.2 Topology in Normed Spaces 184
7.3 Inner Product Spaces 187
7.4 Orthonormal Sets 191
•7.5 Orthogonal Expansions in Inner Product Spaces 196
•7.6 Finite-Dimensional Normed Spaces 204
? 7.7 The L
p
norms 208
8 Limits of Functions 213
8.1 Limits of Functions 213
8.2 Uniform Convergence and Continuity 218
8.3 Uniform Convergence and Integration 220
8.4 Series of Functions 225
•8.5 Power Series 232
? 8.6 Compactness and Subsets of C(K) 239
9 Metric Spaces 246
9.1 Definitions and Examples 246
•9.2 Compact Metric Spaces 250
•9.3 Complete Metric Spaces 254
•9.4 Connectedness 257
? 9.5 Metric Completion 261
? 9.6 The L
p
spaces and Abstract Integration 266
Contents ix
Part B Applications 273
10 Approximation by Polynomials 275
10.1 Taylor Series 275
10.2 How Not to Approximate a Function 285
10.3 Bernstein’s Proof of the Weierstrass Theorem 290
10.4 Accuracy of Approximation 294
•10.5 Existence of Best Approximations 297
•10.6 Characterizing Best Approximations 300
? 10.7 Expansions Using Chebychev Polynomials 306
? 10.8 Splines 314
? 10.9 Uniform Approximation by Splines 322
? 10.10 Appendix: The Stone–Weierstrass Theorem 326
11 Discrete Dynamical Systems 331
11.1 Fixed Points and the Contraction Principle 332
11.2 Newton’s Method 344
•11.3 Orbits of a Dynamical System 349
•11.4 Periodic Points 355
•11.5 Chaotic Systems 362
? 11.6 Topological Conjugacy 370
? 11.7 Iterated Function Systems and Fractals 378
12 Differential Equations 386
12.1 Integral Equations and Contractions 386
12.2 Calculus of Vector-Valued Functions 390
12.3 Differential Equations and Fixed Points 395
12.4 Solutions of Differential Equations 399
12.5 Local Solutions 405
•12.6 Linear Differential Equations 411
•12.7 Perturbation and Stability of DEs 416
? 12.8 Existence without Uniqueness 420
13 Fourier Series and Physics 423
•13.1 The Steady-State Heat Equation 423
13.2 Formal Solution 427
13.3 Orthogonality Relations 429
13.4 Convergence in the Open Disk 432
13.5 The Poisson Formula 435
13.6 Poisson’s Theorem 439
13.7 The Maximum Principle 443
•13.8 The Vibrating String (Formal Solution) 446
•13.9 The Vibrating String (Rigorous Solution) 450
•13.10 Appendix: The Complex Exponential 454
x Contents
14 Fourier Series and Approximation 463
14.1 Least Squares Approximations 463
•14.2 The Isoperimetric Problem 468
14.3 The Riemann–Lebesgue Lemma 471
14.4 Pointwise Convergence of Fourier Series 476
•14.5 Gibbs’s Phenomenon 485
14.6 Ces
`
aro Summation of Fourier Series 488
? 14.7 Best Approximation by Trig Polynomials 495
? 14.8 Connections with Polynomial Approximation 498
? 14.9 Jackson’s Theorem and Bernstein’s Theorem 503
15 Wavelets 513
15.1 Introduction 513
15.2 The Haar Wavelet 515
15.3 Multiresolution Analysis 520
15.4 Recovering the Wavelet 524
•15.5 Daubechies Wavelets 528
•15.6 Existence of the Daubechies Wavelets 534
•15.7 Approximations Using Wavelets 537
? 15.8 The Franklin Wavelet 541
? 15.9 Riesz Multiresolution Analysis 548
16 Convexity and Optimization 557
16.1 Convex Sets 557
16.2 Relative Interior 564
16.3 Separation Theorems 568
16.4 Extreme Points 573
16.5 Convex Functions in One Dimension 576
•16.6 Convex Functions in Higher Dimensions 583
•16.7 Subdifferentials and Directional Derivatives 587
•16.8 Tangent and Normal Cones 596
•16.9 Constrained Minimization 601
•16.10 The Minimax Theorem 608
References 615
Index 617