Table Of ContentSpringer Undergraduate Mathematics Series
Michael Field
Essential
Real Analysis
Springer Undergraduate Mathematics Series
AdvisoryBoard
M.A.J.Chaplain,UniversityofSt.Andrews
A.MacIntyre,QueenMaryUniversityofLondon
S.Scott,King’sCollegeLondon
N.Snashall,UniversityofLeicester
E.Süli,UniversityofOxford
M.R.Tehranchi,UniversityofCambridge
J.F.Toland,UniversityofCambridge
Moreinformationaboutthisseriesathttp://www.springer.com/series/3423
Michael Field
Essential Real Analysis
123
MichaelField
EngineeringMathematicsDepartment
MerchantVenturersSchoolofEngineering
BristolUniversity
UK
DepartmentofMathematics
RiceUniversity
Houston,Texas
USA
ISSN1615-2085 ISSN2197-4144 (electronic)
SpringerUndergraduateMathematicsSeries
ISBN978-3-319-67545-9 ISBN978-3-319-67546-6 (eBook)
https://doi.org/10.1007/978-3-319-67546-6
LibraryofCongressControlNumber:2017955015
Mathematics Subject Classification: 26-01, 40-01, 26Axx, 26Bxx (especially 26B05, 26B10), 26Exx
(especially 26E05, 26E10), 33Bxx (especially 33B15), 34A12, 40Axx, 42Axx (especially 42A10),
54Exx(especially54-01,54E35)
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Preface
This book is an introduction to real analysis: the foundations, the nature of the
subject, and some of the big results. It is based on long personal experience of
teaching undergraduate and graduate level analysis to a diverse range of classes
in England(Warwick),Australia (Sydney)and the United States (Houston)andis
intendedtoappealtomathematicianswitheitherapureorappliedfocus.
Topicsinthebookaredrawnfromseventeenthcenturytolatetwentiethcentury
analysis. While the techniquesof analysis naturally form an importantpart of the
book,theemphasisisonpresentingabroadspectrumofsomeofthepowerfuland
beautiful results that can be proved using analytic methods. Many of the results
are important in applications—for example, Fourier series and asymptotics—and
shouldhelpprovideasoundfoundationforworkinAppliedMathematics.
At the conclusionof the preface,we give a detailed descriptionof the contents
of the book, together with advice for the reader and comments about what is not
included and why. For now, we mention a few of the highlights. We include a
wide range of results on Fourier series. Applications include the infinite product
formula for sin.x/, used in the analysis of the Gamma function, and results on
Bernoullipolynomialsleadingtoexplicitformulasforthesumofanumberofwell-
known infinite series. We give an introduction to the theory of smooth (infinitely
differentiable, non-analytic) functions with an emphasis on how to construct a
smooth function with specific properties. There is an extensive development of
the theoryof metric spaces (includingapplicationsof both the Arzelà–Ascoliand
contraction mapping theorems to differential equations). This is followed by a
short chapter on the Hausdorff metric, which includes a pretty application of the
contractionmappinglemmatofractalgeometryanditeratedfunctionsystems.The
finalchaptercontainsasystematicdevelopmentofthedifferentialcalculusonfinite-
dimensional vector spaces and includes results ranging from the equivalence of
normsonafinite-dimensionalvectorspace,throughtheimplicitfunctionandrank
theorems to a proof of the strong version of the existence and regularity theorem
fordifferentialequations.Theguidingprinciplethroughoutisthatthegeneralityof
conceptsandlanguageintroducedinthetexthastobejustifiedbytheapplication.
Wehavemoretosayaboutthisbelowbut,asanexample,weemphasizesequential
v
vi Preface
compactnessrather than compactness(open cover definition) simply because it is
difficulttofindgoodapplicationsatthislevelthatneedthemoreabstractdefinition
of compactness. In this book, abstraction and generality has to be justified by
applicationand/ortransparency.We abstractas much as we can butno more than
weneedto.
Inthenextfewparagraphs,Isketchsomeofthereasonsandconcernsthatledto
thewritingofthebook.Ihopethesecommentswillhelpbothstudentandinstructor
tomakethebestuseofthetext.
Analysis is a beautiful, powerful and central part of mathematics. Yet I feel
that the way the subject is sometimes presented in undergraduate courses can be
deceptive,misleadinganduninspiring.
Many of the insights, ideas and practices of mathematics are coded into the
language and notation of mathematics. As Alexandre Borovik remarks in his
illuminatingbook:MathematicsundertheMicroscope.NotesonCognitiveAspects
ofMathematicalPractice
‘Mathematicallanguagesunstoppablydeveloptowardsaneverincreasingdegreeofcom-
pressionofinformation’[4,page68].
Howdoesonelearnaforeignlanguage?Itdependsonthecontext.Ifconversa-
tional skills are what is required, then the best way to proceed is to be immersed
in the culture; to be surrounded by people who are native speakers. On the other
hand,ifthegoalistoreadclassicRomanorGreekliterature,thenitisapainstaking
processoflearningthelanguageandgrammarstep-by-stepwiththeeventualaimof
readingVirgilorSophoclesintheoriginal.Immersionintheculturecanbeeffective
andpleasurablewhereaslearningfromthebookcanbeaslowandpainfulprocess
(myownexperiencetranslatingCaesar’sGallicWarIIIfromtheLatin).
Spokenlanguagesevolverelativelyslowly;eventhoughtheseventeenthcentury
plays of Shakespeare can be difficult to read, a production of a Shakespeare is
usuallyeasytounderstand—thoughsomeofthepunsandwordplayscanbemissed.
Learningandreadingmathematicsisatrickierproposition.
Incontrasttospokenlanguage,mathematicallanguagescanevolveandchange
rapidlyand,asBoroviknotes,containincompressedformmuchinformationabout
theideasandpractice.Themodernabstractlanguageofanalysiswouldbehardfor
aseventeenthoreighteenthcenturymathematiciantograsp.Theymightreasonably
well ask, ‘why on earth are you going to so much trouble?’ It is not just the
compressed way definitions are handled; there is all the infrastructure of logic
and set theory that we now take for granted but which was originally regarded
as controversial, even at the beginning of the twentieth century. How then does
language mesh with mathematical content in contemporaryundergraduateclasses
incalculusandanalysis?Insubstance,asopposedtolanguage,muchcontemporary
undergraduate analysis barely goes beyond eighteenth century mathematics. For
example, the one-variable part of a calculus sequence is largely stuck in the
seventeenth century and multivariable calculus rarely gets far into the nineteenth
century.Themathematicallanguageusedinthesecoursescanoftenbeafractured
Preface vii
mix of the old and new which does not resonate well with the content. It may be
helpfultodiscussoneexampleindetail.
The";ı-definitionofcontinuity.Thisisinvariablygivenincalculustexts,usually
as part of a discussion on limits. Subsequently,the topic is largely ignored—with
proofs or argumentsdeferred as being too difficult. Versions of the ";ı-definition
of continuitywere originallyproposed by Bernard Bolzano (1817)and Augustin-
LouisCauchy(about1820)andthenputonamoresoundand,tooureyes,familiar
footingbyKarlWeierstrassabout20yearslater.BolzanoandCauchywereamong
the first mathematicians to stress the importance of rigour in analysis. Even so,
Cauchy’s original use of the continuity definition was incorrect: he argued that a
pointwise limit of a sequence of continuousfunctionswas continuous.1 Although
questionsaboutthe meaning and nature of limits, especially associated to tangent
lines,haveplayedaroleinanalysissincethetimeofLeibnizandNewton,theidea
ofacontinuousfunctionisrelativelyrecent,spurredinpartbytheearlynineteenth
century developmentof Fourier series. Continuity plays a peripheralrole in most
contemporarycalculusclasses and evenin introductoryanalysisclasses it is often
poorlymotivated—itis quitepossibleto completea mathematicsdegreeandhave
little, if any, contact with Fourier series or functions outside of combinations of
polynomials,trigonometricandexponentialfunctions.
The ";ı-definition of continuity is difficult: a continuous function is not quite
whatyouthinkitis—atypicalcontinuousfunctionisnowheredifferentiable.Likein
muchofanalysis,thedefinitionisframedintermsofinequality(tricky)ratherthan
theeasierequalityseeninelementaryalgebra.Frankly,thecontinuitydefinitionis
ugly and uses too many quantifiers (for all x, for all " > 0, there exists...). In
the twentiethcentury,a morenaturaltopologicaldefinitionof continuityappeared
whichisbasedonpreservationofstructure(opensets).Thedefinitionisfarreaching
andappliestomanyareasofmathematics,includingalgebra,butunfortunatelythe
definitionisevenmoreremotefromthecontentofastandardcalculussequence.
Can continuity be motivated in a calculus or introductory analysis course? In
particular, are there significant results or examples? One possible approach is to
prove,oratleastdiscuss,theuniquenessandexistenceoftheintegralofacontinuous
function.
In a calculus class, the word ‘integral’ is, in practice, synonymouswith ‘anti-
derivative’.Itiseasytoshowrigorouslythatiftheintegralofacontinuousbounded
functionexists,thenitmustbegivenbytheanti-derivative(thisisuniquenessofthe
integral).Conversely,theproofofexistenceoftheanti-derivativeforacontinuous
boundedfunctionisnothard(theproofdoesnotrequireuniformcontinuity).While
existencedoesdependoncompletenesspropertiesoftherealnumbers(asissofor
mostresultsinrealanalysis),itdoesnotdependongeneralpropertiesofcontinuous
functions (for example, that a continuous function on a closed interval attains its
bounds). In spite of the simplicity of proving the existence of the integral of a
1ThecorrectresultwasobtainedlaterbyWeierstrassusingtheideaofuniformconvergence.This,
andalltheotherresultsanddefinitionswemention,canbefoundinthemainbodyofthetext.
viii Preface
continuousboundedfunction,calculustextspersistingivingadevelopmentbased
on upper and lower Riemann sums (so presenting the theory as an extension of
themethodofexhaustiondevelopedbyEudoxusandlaterArchimedes)andalmost
universallyclaimtheproofofexistenceistoodifficulttoinclude.Itisnot.(Werefer
to Chap.2, Sect. 2.8 for a simple presentation of the existence and uniquenessof
theintegralofacontinuousboundedfunction.)Thewholepointabouttheintegral
ofacontinuousboundedfunctionisthatonedoesnotneedtodoapproximationby
Riemannsumsifthefunctionhasananti-derivative.Thisispartofthemagicofthe
integralanddifferentialcalculus.2
In practice, calculus and introductory analysis courses only consider functions
constructed from polynomials, trigonometric, exponential and logarithmic func-
tions.Forthesefunctions,itisusuallyeasyto proveboundednessona closedand
bounded interval. The differential calculus can then be used to find the bounds
and where they are attained. There is no real connection between the abstract
theoryofcontinuousfunctionsonaclosedandboundedintervalandthefunctions
consideredinacalculusclasswhichareinvariablyanalytic(inparticular,infinitely
differentiable). What is interesting and remarkable is the existence theorem that
everycontinuousfunction,suchascos.x2/,doeshaveananti-derivativeeventhough
it cannot always be given in closed form (in terms of combinations of known
functions).Theseissuesareoftennotdiscussedinmoderntextbooks.Insummary,
highly sophisticated definitions and language are introduced for the solution of
problemsthatarenevermentioned.Itisnotsurprisingthatstudentscanbebaffled
bycalculusclasses:thereisadisconnectbetweenthematerialandthelanguage.
These problems can be reinforced when students take an introductory class
on analysis. These courses are often presented3 as ‘an introduction to proof’ or
‘calculus done properly’. Aside from the poor psychology (‘what you did before
was wrong and so a waste of time’), the premise is wrong: this abstraction is not
neededtodocalculusormuchofclassicalanalysis(asistestifiedtobytheworkof
Eulerandothereighteenthcenturymathematicians).
Amongstmathematiciansthere arestronglyheld viewsaboutteachingstudents
how to write proofs. My personal view is that it is inadvisable to overemphasize
the nuances of logic, truth tables, existential quantifiers etc. Often this leads to
overuse of symbols combined with a lack of understanding of the underlying
mathematics. Symbols seem powerful. Even though (perhaps because) I was
brought up on Axiomatic Euclidean Geometry (the mathematical equivalent of
learning Latin), I am sceptical about doing serious mathematics in the axiomatic-
didactic style. It confuses the process of doing mathematics with the activity of
writing mathematics. Figuringout a proofis an intuitive processthat results from
increasingunderstandingofthemathematicalstructure;formalismcomesin when
2The interest of the Riemann sum approach to the integral is that the construction works for
boundedfunctionswhichhavecountablymanydiscontinuities.Inthatcontext,upperandlower
sumsareneeded.Thiswashighlyrelevantfornineteenthcenturymathematics.
3IntheUnitedStates.
Preface ix
one writes the proof down (in a terse coded form) so that others can understand
and use the result. Introductoryanalysis presents rich problemsthat can naturally
leadtothelogicalstyleoneneedsforwritingproofs(forexample,theproofthata
sequentially continuous function is continuous). An essential part of this process
is developing the skill to construct good examples and counterexamples. In my
view,thewaytolearnlogicandmathematicalexpressionisthroughapplicationnot
throughaformalcourse.Emphasissolelyoncorrectnessisantitheticaltodeveloping
theintuitiveunderstandingthatoneneedstodomathematics.
There can also be disconnects between language and application in more
advancedundergraduateclasses on analysis. Metricspaces area terrificsubjectto
learnatanylevel.Thereisgeometricintuitioncombinedwithmanypowerfulresults
(notablythe contractionmappinglemma) and goodapplications.The subject also
providesa beautifulandeasily accessible abstraction,generalizationand clarifica-
tionofmuchfoundationalonevariablerealanalysis.Definitionsandresultscanbe
givenusingelementarysequence-baseddefinitionssuchassequentialcontinuityor
sequentialcompactness.Thesedefinitionsleadtosimpleandtransparentproofs.In
contrast, if one emphasizes the topologicalapproach, for example the open cover
definition of compactness, proofs are not so transparent and often harder.4 Most
function spaces encountered in elementary analysis are separable metric spaces
where sequence-based methods work very well. At undergraduate level, it is not
so easy to give interesting examples of non-separable or non-metrizable spaces.5
QuotingBorovikagain
‘Always test a mathematical theory on the simplest possible example—and explore the
exampletotheutmostlimits’[4,Page3].
As the late Christopher Zeeman said, ‘a good example is worth 10 theorems’.
Counterexamples are important too—as a way of testing limits of the theory (as
well as the need for the theory).6 In summary, and paraphrasing Albert Einstein,
‘Everythingshouldbemadeassimpleaspossible,butnosimpler.’Ifthereisagood
reasontoconsiderthetopologyofnon-separableornon-metrizablemetricspacesin
thecourse,thengivethetopologicaldefinitionofcompactness.Else,keepitsimple.
One reasongivento stress formoverapplicationina finalyearanalysiscourse
is that the course should be preparation for graduate classes in analysis. I feel
this approachis mistaken. As a professionalmathematician, I should be inspiring
undergraduatestudentsaboutthenatureandpowerofmathematicsandnotcovering
the preliminaries for a possible future graduate class. A final year undergraduate
courseshouldbecompleteinitselfandcontaininterestingandexcitingapplications.
Ifthisisnotdone,itislikelearningLatintoreadVirgilbutneveractuallyreading
4Asimpleexampleisgivenbytheproofofuniformcontinuityofacontinuousfunctiononaclosed
andboundedinterval.
5Spaces of functions of bounded variation are metrizable but not separable. Spaces of smooth
functionsonRwiththeWhitneyC1topologyareneitherseparablenormetrizable.
6Examplesshowingthatresultsonboundednessofcontinuousfunctionsonclosedintervalsfailif
weworkovertherationalnumbers.
