Counterexamples on Uniform Convergence: Sequences, Series, Functions, and Integrals

273 Pages·2017·4.087 MB·English

by  Andrei Bourchtein, Ludmila Bourchtein

CounterexamplesonUniformConvergence

Counterexamples on Uniform Convergence
Sequences,Series,Functions,andIntegrals
AndreiBourchtein
LudmilaBourchtein
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LibraryofCongressCataloging-in-PublicationData
Names:Bourchtein,Andrei.|Bourchtein,Ludmila.
Title:Counterexamplesonuniformconvergence:sequences,series,functions,andintegrals/
AndreiBourchtein,LudmilaBourchtein.
Description:Hoboken,NewJersey:JohnWiley&Sons,Inc.,c2017.|
Includesbibliographicalreferencesandindex.
Identifiers:LCCN2016038687|ISBN9781119303381(cloth)|ISBN9781119303428(epub)|
ISBN9781119303404(epdf)
Subjects:LCSH:Mathematicalanalysis–Problems,exercises,etc.|
Calculus–Problems,exercises,etc.|Sequences(Mathematics)|Functions.|Integrals.
Classification:LCCQA301.B682017|DDC515–dc23
LCrecordavailableathttps://lccn.loc.gov/2016038687
CoverImagecredit:Abscent84/gettyimages
Setin10/12ptWarnockbySPiGlobal,Pondicherry,India
PrintedintheUnitedStatesofAmerica
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vii
Contents
Preface ix
ListofExamples xi
ListofFigures xxix
AbouttheCompanionWebsite xxxiii
Introduction xxxv
I.1 Comments xxxv
I.1.1 OntheStructureofThisBook xxxv
I.1.2 OnMathematicalLanguageandNotation xxxvii
I.2 Background(ElementsofTheory) xxxviii
I.2.1 SequencesofFunctions xxxviii
I.2.2 SeriesofFunctions xli
I.2.3 FamiliesofFunctions xliv
1 ConditionsofUniformConvergence 1
1.1 Pointwise,Absolute,andUniformConvergence.Convergenceona
SetandSubset 1
1.2 UniformConvergenceofSequencesandSeriesofSquaresand
Products 15
1.3 Dirichlet’sandAbel’sTheorems 31
Exercises 39
FurtherReading 42
2 PropertiesoftheLimitFunction:Boundedness,Limits,
Continuity 45
2.1 ConvergenceandBoundedness 45
2.2 LimitsandContinuityofLimitFunctions 51
2.3 ConditionsofUniformConvergence.Dini’sTheorem 68
2.4 ConvergenceandUniformContinuity 79
Exercises 88
FurtherReading 93
viii Contents
3 PropertiesoftheLimitFunction:Differentiabilityand
Integrability 95
3.1 DifferentiabilityoftheLimitFunction 95
3.2 IntegrabilityoftheLimitFunction 117
Exercises 128
FurtherReading 131
4 IntegralsDependingonaParameter 133
4.1 ExistenceoftheLimitandContinuity 133
4.2 Differentiability 144
4.3 Integrability 154
Exercises 162
FurtherReading 166
5 ImproperIntegralsDependingonaParameter 167
5.1 Pointwise,Absolute,andUniformConvergence 167
5.2 ConvergenceoftheSumandProduct 176
5.3 Dirichlet’sandAbel’sTheorems 185
5.4 ExistenceoftheLimitandContinuity 192
5.5 Differentiability 198
5.6 Integrability 202
Exercises 210
FurtherReading 214
Bibliography 215
Index 217