Basic real analysis

687 Pages·2014·3.36 MB·English

by  Houshang H. Sohrab

Houshang H. Sohrab
Basic Real
Analysis
Second Edition

Houshang H. Sohrab
Basic Real Analysis
Second Edition
HoushangH.Sohrab
Mathematics
TowsonUniversity
Towson,MD,USA
ISBN978-1-4939-1840-9 ISBN978-1-4939-1841-6(eBook)
DOI10.1007/978-1-4939-1841-6
SpringerNewYorkHeidelbergDordrechtLondon
LibraryofCongressControlNumber:2014949237
MathematicsSubjectClassification(2010):26-01,26Axx,28-01,40-01,46-01,54-01,60-01
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ToShohreh,Mahsa,andZubin

Preface
Ah,Love!couldthouandIwithFateconspire
TograspthissorrySchemeofThingsentire,
Wouldnotweshatterittobits—andthen
Re-moulditnearertotheHeart’sDesire!
OmarKhayyam,Rubaiyat
More than 10years have passed since the publication of the first edition of this
textbook.Duringtheseyears,alargenumberofmonographsdealingwiththesame
topics have appeared. Some of them have been included in the new bibliography.
In addition, a wealth of material is now freely available online, some of it posted
bytheverybest(cf.,e.g.,[Tao11]).Soonemayquestionthewisdomofofferinga
new edition of the old Basic Real Analysis, henceforth abbreviated BRA. And yet,
as is always the case, different people look at the same material in different ways
dependingontheirtastes.Whatshouldorshouldnotbeincludedandtowhatextent
mayvaryconsiderably,andallchoiceshavetheirlegitimateandlogicaljustification.
Despite the fact that I have looked at a large number of real analysis textbooks
andhavebenefitedfromallofthem,Istillprefernottomodifytheorganizationof
the material in BRA. The initial idea of a new edition came from Tom Grasso of
Birkhäuser, and I want to use this opportunity to thank him for suggesting it. He
pointedoutthatfortheprojecttobejustified,areasonablenumberofchangesmust
bemade.ThemostsubstantialchangeintheneweditionisthatIrewroteChaps.10
and11onLebesguemeasureandintegralentirely.Indoingso,Idecidedtoabandon
F.Riesz’smethodusedinthefirsteditioninfavorofthemoretraditionalapproach
oftreatingLebesguemeasurebeforeintroducingtheintegral.Ihavecometobelieve
that measuretheory isafundamental part ofanalysis andthesooner one learnsit,
thebetter.
Lebesgue measure and integral on the real line are now covered in Chap.10.
Chapter11containsadditionaltopics,includingaquicklookatimproperRiemann
integrals,integralsdependingonaparameter,theclassicalLp-spaces,othermodes
ofconvergence,andafinalsectiononthedifferentiationproblem.Thislastsection
containsLebesgue’stheoremonthedifferentiabilityofmonotonefunctions(withF.
Riesz’sRisingSunLemmausedintheproof)andhisversionsoftheFundamental
Theorem(s) of Calculus. Abstract measure and integration are treated in Chap.12,
whereIhaveincludedtheRadon–Nikodymtheoremwhichisusedinthelastsection
onprobability.
vii
viii Preface
Although thenewly writtenchapters onLebesgue’s theory constitutethemajor
change in this edition, all other chapters have been affected to various degrees.
Forexample,thetreatmentofconvexfunctionshasbeenmodifiedand(hopefully)
improved. Ihaveaddedanumberofexercises inthetextandmanynew problems
attheendofallchapters.Alargenumberoftypographicalaswellasmoreserious
errorshavebeencorrected.IamparticularlyindebtedtoProfessorGiorgioGiorgiof
UniversitàDegliStudiDiPaviaforpointingoutaseriousone.Ofcourse,asalways,
other undetected errors may still be there and I take full responsibility for them.
Needlesstosay,Iwouldbegratefultocarefulreadersforpointingthemouttome
([email protected]).
Ideally, a book at this level should include some spectral theory, say, at least
thespectralpropertiesofcompact,self-adjointoperators.Unfortunately,thiswould
increase the size of the book beyond what I consider to be reasonable. I have
decidedtoincludesomeofthisandsimilarinterestingmaterialintheend-of-chapter
problems, and the interested readers may try as many of them as they want. A
complete solution manual is available from the publisher for the benefit of the
potential instructors. I have decided to use sequential numbering of all the items
throughout.Ibelievethatthissimplifiesthenavigationconsiderablyeventhoughit
mayhaveitsproblems.
It is a great pleasure to thank Mitch Moulton, Birkhäuser’s assistant editor, for
hishelpandpatienceduringthepreparationofthemanuscript.Iamalsogratefulfor
thetechnicalassistanceIreceivedfromBirkhäuser.OneofthepeopleIcompletely
forgot to thank in the first edition of BRA (shame on me!) is Loren Spice. He was
16 when he started enrolling in mathematics courses at Towson University, right
whenIwaspreparingthefirstdraftofthetextbook. Hereadthefirstfivechapters
verythoroughlyandmadealargenumberofsuggestionsandcorrections.Iamtruly
indebtedtohimforhisvaluablecommentswhichresultedinmanyimprovements.
Also,Iowesomuchtothebrilliant,anonymousreviewerofthefirsteditionofBRA
whose excellent critical comments had a great influence, even though I couldn’t
possibly live up to his high expectations. I hope he finds this new edition to be
closer to his taste. In addition, the anonymous reviewers of this new edition have
madeanumberofexcellentcommentsforwhichIamtrulygrateful.
Finally, I would like to thank my wife, Shohreh, and my children, Mahsa and
Zubin,whoseloveandsupportarethegreatestdrivingforceinmylife.
Towson,MD,USA HoushangSohrab
Contents
1 SetTheory................................................................... 1
1.1 RingsandAlgebrasofSets .......................................... 1
1.2 RelationsandFunctions.............................................. 6
1.3 BasicAlgebra,Counting,andArithmetic........................... 17
1.4 InfiniteDirectProducts,AxiomofChoice,andCardinal
Numbers .............................................................. 29
1.5 Problems.............................................................. 34
2 SequencesandSeriesofRealNumbers .................................. 39
2.1 RealNumbers ........................................................ 39
2.2 SequencesinR....................................................... 50
2.3 InfiniteSeries......................................................... 61
2.4 UnorderedSeriesandSummability ................................. 78
2.5 Problems.............................................................. 86
3 LimitsofFunctions......................................................... 97
3.1 BoundedandMonotoneFunctions.................................. 97
3.2 LimitsofFunctions .................................................. 100
3.3 PropertiesofLimits .................................................. 102
3.4 One-SidedLimitsandLimitsInvolvingInfinity.................... 107
3.5 IndeterminateForms,Equivalence,andLandau’sLittle
“oh”andBig“Oh” ................................................... 117
3.6 Problems.............................................................. 125
4 TopologyofRandContinuity............................................. 129
4.1 CompactandConnectedSubsetsofR .............................. 130
4.2 TheCantorSet........................................................ 134
4.3 ContinuousFunctions................................................ 140
4.4 One-SidedContinuity,Discontinuity,andMonotonicity........... 147
4.5 ExtremeValueandIntermediateValueTheorems.................. 153
4.6 UniformContinuity .................................................. 159
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