Real Analysis with Real Applications

642 Pages·2012·3.51 MB·English

Real Analysis
with
Real Applications

Real Analysis
with
Real Applications
Kenneth R. Davidson
UniversityofWaterloo
Allan P. Donsig
UniversityofNebraska
PrenticeHall
UpperSaddleRiver,NJ07458
LibraryofCongressCataloging–in–PublicationData
Davidson,KennethR.
Realanalysiswithrealapplications/KennethR.Davidson,AllanP.Donsig.
p.cm.
Includesbibliographicalreferencesandindex.
ISBN0-13-041647-9
1.Mathematicalanalysis.I.Donsig,AllanP.II.Title.
QA300.D3422002
515–dc21
2001052318
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C
ONTENTS
Preface xi
DependenceonEarlierSections xiv
PossibleCourseOutlines xvi
1 Background 1
1.1 TheLanguageofMathematics 1
1.2 SetsandFunctions 5
1.3 Calculus 10
1.4 LinearAlgebra 11
1.5 TheRoleofProofs 19
1.6 Appendix: EquivalenceRelations 25
†
Part A Abstract Analysis
29
2 The Real Numbers 31
2.1 AnOverviewoftheRealNumbers 31
2.2 InfiniteDecimals 34
2.3 Limits 37
2.4 BasicPropertiesofLimits 42
2.5 UpperandLowerBounds 46
2.6 Subsequences 51
2.7 CauchySequences 55
2.8 Appendix: Cardinality 60
†
3 Series 66
3.1 ConvergentSeries 66
3.2 ConvergenceTestsforSeries 70
3.3 TheNumbere 77
†
3.4 AbsoluteandConditionalConvergence 80
†
4 The Topology of Rn 88
4.1 n-dimensionalSpace 88
4.2 ConvergenceandCompletenessinRn 92
4.3 ClosedandOpenSubsetsofRn 96
4.4 CompactSetsandtheHeine–Borel Theorem 101
vii
viii Contents
5 Functions 108
5.1 LimitsandContinuity 108
5.2 DiscontinuousFunctions 114
5.3 PropertiesofContinuousFunctions 120
5.4 CompactnessandExtremeValues 124
5.5 UniformContinuity 127
5.6 TheIntermediateValueTheorem 133
5.7 MonotoneFunctions 135
†
6 Differentiation and Integration 141
6.1 DifferentiableFunctions 141
6.2 TheMeanValueTheorem 148
6.3 RiemannIntegration 153
6.4 TheFundamentalTheoremofCalculus 164
6.5 Wallis’sProductandStirling’sFormula 169
†
?6.6 MeasureZeroandLebesgue’sTheorem 175
7 Normed Vector Spaces 179
7.1 DefinitionandExamples 179
7.2 TopologyinNormedSpaces 184
7.3 InnerProductSpaces 187
7.4 OrthonormalSets 191
7.5 OrthogonalExpansionsinInnerProductSpaces 196
†
7.6 Finite-DimensionalNormedSpaces 204
†?7.7 TheLp norms 208
8 Limits of Functions 213
8.1 LimitsofFunctions 213
8.2 UniformConvergenceandContinuity 218
8.3 UniformConvergenceandIntegration 220
8.4 SeriesofFunctions 225
8.5 PowerSeries 232
†
?8.6 CompactnessandSubsetsofC(K) 239
9 Metric Spaces 246
9.1 DefinitionsandExamples 246
9.2 CompactMetricSpaces 250
†
9.3 CompleteMetricSpaces 254
†
9.4 Connectedness 257
†
?9.5 MetricCompletion 261
?9.6 TheLp spacesandAbstractIntegration 266
Contents ix
Part B Applications
273
10 Approximation by Polynomials 275
10.1 TaylorSeries 275
10.2 HowNottoApproximateaFunction 285
10.3 Bernstein’sProofoftheWeierstrassTheorem 290
10.4 AccuracyofApproximation 294
10.5 ExistenceofBestApproximations 297
†
10.6 CharacterizingBestApproximations 300
†
?10.7 ExpansionsUsingChebychevPolynomials 306
?10.8 Splines 314
?10.9 UniformApproximationbySplines 322
?10.10 Appendix: TheStone–WeierstrassTheorem 326
11 Discrete Dynamical Systems 331
11.1 FixedPointsandtheContractionPrinciple 332
11.2 Newton’sMethod 344
11.3 OrbitsofaDynamicalSystem 349
†
11.4 PeriodicPoints 355
†
11.5 ChaoticSystems 362
†
?11.6 TopologicalConjugacy 370
?11.7 IteratedFunctionSystemsandFractals 378
12 Differential Equations 386
12.1 IntegralEquationsandContractions 386
12.2 CalculusofVector-ValuedFunctions 390
12.3 DifferentialEquationsandFixedPoints 395
12.4 SolutionsofDifferentialEquations 399
12.5 LocalSolutions 405
12.6 LinearDifferentialEquations 411
†
12.7 PerturbationandStabilityofDEs 416
†
?12.8 ExistencewithoutUniqueness 420
13 Fourier Series and Physics 423
13.1 TheSteady-StateHeatEquation 423
†
13.2 FormalSolution 427
13.3 OrthogonalityRelations 429
13.4 ConvergenceintheOpenDisk 432
13.5 ThePoissonFormula 435
13.6 Poisson’sTheorem 439
13.7 TheMaximumPrinciple 443
13.8 TheVibratingString(FormalSolution) 446
†
13.9 TheVibratingString(RigorousSolution) 450
†
13.10 Appendix: TheComplexExponential 454
†
x Contents
14 Fourier Series and Approximation 463
14.1 LeastSquaresApproximations 463
14.2 TheIsoperimetricProblem 468
†
14.3 TheRiemann–Lebesgue Lemma 471
14.4 PointwiseConvergenceofFourierSeries 476
14.5 Gibbs’sPhenomenon 485
†
14.6 Cesa`roSummationofFourierSeries 488
?14.7 BestApproximationbyTrigPolynomials 495
?14.8 ConnectionswithPolynomialApproximation 498
?14.9 Jackson’sTheoremandBernstein’sTheorem 503
15 Wavelets 513
15.1 Introduction 513
15.2 TheHaarWavelet 515
15.3 MultiresolutionAnalysis 520
15.4 RecoveringtheWavelet 524
15.5 DaubechiesWavelets 528
†
15.6 ExistenceoftheDaubechiesWavelets 534
†
15.7 ApproximationsUsingWavelets 537
†
?15.8 TheFranklinWavelet 541
?15.9 RieszMultiresolutionAnalysis 548
16 Convexity and Optimization 557
16.1 ConvexSets 557
16.2 RelativeInterior 564
16.3 SeparationTheorems 568
16.4 ExtremePoints 573
16.5 ConvexFunctionsinOneDimension 576
16.6 ConvexFunctionsinHigherDimensions 583
†
16.7 SubdifferentialsandDirectionalDerivatives 587
†
16.8 TangentandNormalCones 596
†
16.9 ConstrainedMinimization 601
†
16.10 TheMinimaxTheorem 608
†
References 615
Index 617