分析入门

图书标准信息

• 作者 [美]雅培（Abbott S.） 著
• 出版社 世界图书出版公司
• 出版时间 2008-10
• 版次 1
• ISBN 9787506292795
• 定价 39.00元
• 装帧 平装
• 开本 24开
• 纸张 胶版纸
• 页数 257页
• 正文语种 英语

【内容简介】

MyprimarygoalinwritingUnderstandingAnalysiswastocreateanelementaryone-semesterbookthatexposesstudentstotherichrewardsinherentintakingamathematicallyrigorousapproachtothestudyoffunctionsofarealvariable.Theaimofacourseinrealanalysisshouldbetochallengeandimprovemathematicalintuitionratherthantoverifyit.Thereisatendency,however,tocenteranintroductorycoursetoocloselyaroundthefamiliartheoremsofthestandardcalculussequence.Producingarigorousargumentthatpolynomialsarecontinuousisgoodevidenceforawell-chosendefinitionofcontinuity,butitisnotthereasonthesubjectwascreatedandcertainlynotthereasonitshouldberequiredstudy.Byshiftingthefocustotopicswhereanuntrainedintuitionisseverelydisadvantaged（e.g.,rearrangementsofinfiniteseries,nowhere-differentiablecontinuousfunctions,Fourierseries）,myintentistorestoreanintellectuallivelinesstothiscoursebyofferingthebeginningstudentaccesstosometrulysignificantachievementsofthesubject.

【目录】

Preface
1TheRealNumbers
1.1Discussion：TheIrrationalityof1.414
1.2SomePreliminaries
1.3TheAxiomofCompleteness
1.4ConsequencesofCompleteness
1.5Cantor'sTheorem
1.6Epilogue

2SequencesandSeries
2.1Discussion：RearrangementsofInfiniteSeries
2.2TheLimitofaSequence
2.3TheAlgebraicandOrderLimitTheorems
2.4TheMonotoneConvergenceTheoremandaFirstLookatInfiniteSeries
2.5SubsequencesandtheBolzano-WeierstrassTheorem
2.6TheCauchyCriterion
2.7PropertiesofInfiniteSeries
2.8DoubleSummationsandProductsofInfiniteSeries
2.9Epilogue

3BasicTopologyofR
3.1Discussion：TheCantorSet
3.2OpenandClosedSets
3.3CompactSets
3.4PerfectSetsandConnectedSets
3.5Baire'sTheorem
3.6Epilogue

4FunctionalLimitsandContinuity
4.1Discussion：ExamplesofDirichletandThomae
4.2FunctionalLimits
4.3CombinationsofContinuousFunctions
4.4ContinuousFunctionsonCompactSets
4.5TheIntermediateValueTheorem
4.6SetsofDiscontinuity
4.7Epilogue

5TheDerivative
5.1Discussion：AreDerivativesContinuous?
5.2DerivativesandtheIntermediateValueProperty
5.3TheMeanValueTheorem
5.4AContinuousNowhere-DifferentiableFunCtion
5.5Epilogue

6SequencesandSeriesofFunctions
6.1Discussion：BranchingProcesses
6.2UniformConvergenceofaSequenceofFunctions
6.3UniformConvergenceandDifferentiation
6.4SeriesofFunctions
6.5PowerSeries
6.6TaylorSeries
6.7Epilogue

7TheRiemannIntegral
7.1Discussion：HowShouldIntegrationbeDefined?
7.2TheDefinitionoftheRiemannIntegral
7.3IntegratingFunctionswithDiscontinuities
7.4PropertiesoftheIntegral
7.5TheFundamentalTheoremofCalculus
7.6Lebesgue'sCriterionforRiemannIntegrability
7.7Epilogue

8AdditionalTopics
8.1TheGeneralizedRiemannIntegral
8.2MetricSpacesandtheBaireCategoryTheorem
8.3FourierSeries
8.4AConstructionofRFromQ
Bibliography
Index