Real Analysis for the Undergraduate: With an Invitation to Functional Analysis

423 Pages·2014·4.39 MB·English

by  Matthew A. Pons (auth.)

Matthew A. Pons
Real Analysis
for the
Undergraduate
With an Invitation to Functional
Analysis
Real Analysis for the Undergraduate

Matthew A. Pons
Real Analysis for the
Undergraduate
With an Invitation to Functional Analysis
with Illustrations by Robert F. Allen
123
MatthewA.Pons
NorthCentralCollege
Naperville,IL,USA
ISBN978-1-4614-9637-3 ISBN978-1-4614-9638-0(eBook)
DOI10.1007/978-1-4614-9638-0
SpringerNewYorkHeidelbergDordrechtLondon
LibraryofCongressControlNumber:2013956133
MathematicsSubjectClassification(2010):26-01
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Preface
Likemanymathematicaltextbooks,thisprojectevolvedfromasetofcoursenotes.
These notes were developed and refined over a 5-year period and represent my
interpretation of a first course in analysis. Much of the material comes from the
standard undergraduate canon including an axiomatic or constructive exploration
of the real number system, numerical sequences and series, limits and continuity,
differentiation, sequences and series of functions, and some introductory form
of integration. In addition I have included material on measure theory and the
Lebesgue integral, and a brief invitation to functional analysis. Although these
advanced topics are not typically introduced until graduate study, I find that with
the proper presentation they are perfectly manageable for undergraduates wishing
todelvedeeperintothesubjectmatter.Myprimaryreasonfortheinclusionofthis
advanced material is to prepare students for graduate study, and to also help them
experiencetheevolutionofmathematics.Whileeachchaptercoversatopiccentral
to a beginning course in real analysis, the last section in each chapter introduces
a topic from functional analysis which is derived from the core chapter content
in a natural way; these sections are developed alongside a concrete topic in order
to provide grounding. This takes the shape of a particular idea, complete normed
linear spaces, eigenvalues, invariant subspaces, or focuses on a particular example
or class of examples, `p spaces, continuous function spaces, and Lp spaces over
generalmeasurespaces.Takenasawhole,thesesectionsprovideaniceconnection
tolinearalgebraandpointtoathrivingareaofmathematicalresearch.TheEpilogue
thenprovidesabriefroadmaptofurtherpointsofinterestintherealmoffunctional
analysis.
When I began this project, I first mapped out the sections within each chapter.
This led to nine chapters which built on each other, but seemed to lack a sense of
harmony.Asareworking,Ibegantothinkabouttheoverallstorylineofthetext,of
eachchapter,andthenofeachsection.Thegoalwastotiethesestorylinestogether
inacohesivemannerthatconveysthestoryofrealanalysis.Thisactivityledtofive
corebeliefswhichhaveinfluencedeverywordwrittenhere.
vii
viii Preface
• An introductory text should be written for students. There are a great many
analysis texts available on the market. Some classical texts are too difficult for
today’s audience (which has changed drastically in the last few decades) and
many of the newer expositions are too watered down. Others seem to strike a
balancebetweenrigorousexpositionandintuitivereasoning,butfallshortwhen
it comes to adequate exercises and examples. In thinking about all of this, my
goal in writing this text was to provide a thorough treatment of the necessary
subject matter without sacrificing rigor, while shaping the presentation so that
every student required to take a course in real analysis can succeed. Examples
arefrequent,discussionofnewideasmotivatesdefinitions,theorems,andproofs
before formal statements and arguments are given, and the exercise sets are
detailed and thorough. The discussion portions of the text are meant for the
newcomer andtherewillinevitablybeinstanceswhereaninstructorthinksI’m
belaboring a point. And while that may be the case, these explanations have
arisenoutofconversationswithstudentsaboutthematerial,bothintheclassroom
andintheofficehours.Undergraduatesthinkdifferentlyaboutnewmaterialthan
aprofessionalmathematicianandmyaudienceiscomprisedofundergraduates.
• Afirstcourseinanalysisshould,tosomedegree,focusonprovidingatheoretical
treatmentofthematerialfromthefirst-yearcalculussequence.Iseethiscourseas
ameansofcomingfullcircle.Studentshaveexperiencedcalculus,theirintuition
has been developed, and they have been exposed to the many applications of
differentiationandintegration.Tocompletethisexperience,itisnecessarythen
that they should understand why the things they have taken on faith are in fact
true. To do this requires a thorough study of the concept of a limit, and it is
necessarythattheyunderstandwhyitistheunderlyingconstructforeverytopic
inanalysis.ThiscantakemanyformsandIhavechosentofocusonsequential
limits.So,whileIdostrivetoprovideaconcretefoundationforthedifferential
andintegralcalculus,Idonotbelievethatatextshouldbeconfinedtodeveloping
this material in the same manner or order as it is presented in the first-year
calculussequence.
• Examples and exercises are the keys to understanding theory. The sections are
writtensoastoprovidemostofthedetailsfortheideasunderconsideration,but
the reader is encouraged time and time again to pick up a pencil and check a
calculation or complete the details of an argument. That being said, core ideas
are spelled out in detail and major results are proved in fullgenerality. Itis my
strongbeliefthatcoreresultsshouldnotbeleftasexercisesasthesearetypically
toodifficultformanystudentsatthebeginnerlevel.
• Studentsdeservetoseehowtheirmathematicscoursesrelatetoeachother;this
in turn leads to an understanding of how mathematics evolves as a discipline.
Mathematics is at its most powerful when it incorporates knowledge from its
various branches, and I find analysis to be particularly rich in this area as it
depends so heavily on the algebra, topology, and geometry of the underlying
domainspace.ThestudyofoperatorsonaHilbertspacemaybethemostideal
exampleofsuchalushandvibrantareaofinquiry.
Preface ix
• Thestoryofanalysisisnotover,soatextshouldnotattempttowrapitupina
tidypackage.Thisagainiswheretheintroductionoffunctionalanalysiscomes
intoplay.Thefunctionaltopicsincludedareageneralizationofthebasicanalysis
topicstheyaccompanyandthesesectionsaredesignedtoproducemorequestions
thantheynecessarilyanswer.Thisisthebeautyofmathematicsandanyacademic
discipline. There will always be questions to ask and my point here is to help
the students see that new questions are being asked not only about new topics
butaboutoldtopicsaswell.Whentakenasawhole,theseculminatingsections
provideabasicintroductiontoveryabstractmaterial.
As for prerequisites, the text assumes that students have experienced calculus
up through sequences and series, which typically comes after differentiation and
integration, and that they have had a formal course in proof writing, including
basiclogicandsettheory,prooftechniques(induction,proofbycontradiction,and
proof by contrapositive), basic function theory, cardinality, and some exposure to
thenatureofquantifiers.Thereareseveralinstanceswherebasictopicsfromlinear
algebra come into play, but these are only present in the culminating sections of
eachchapteranddonotaffectthecorematerialofthechapters;thesecanbeeasily
supplementedtothestudentwhohasnothadaformallinearalgebracourse.
Tothe Instructor
This text is designed for use over the course of two semesters or quarters. I
currently teach the majority of the material over two 10-week terms covering
roughly Chaps.1–5 and 6–9 in the two respective sessions. As mentioned earlier,
the text is written so that each chapter culminates with the introduction of a topic
fromfunctionalanalysissothatthestudentscanseehowthetopicsevolvefromthe
contentofthatparticularchapter.However,Iwillbethefirsttoadmitthatthenatural
orderwillbestrenuousformoststudents.Topreparethestudentsforthechallenge
Iwalkasmootherpath.
I. FirstTerm:Chapters1–5excludingthelastsectionineachchapter;
II. SecondTerm:
a. Chapters6–7excludingthelastsectionineachchapter;
b. Sections1.5,2.5,3.4,6.5,7.5;
c. Chapters8–9;
d. Sections4.5and5.6(iftimeallows).
For a particularly strong group or as an independent reading project, the natural
ordermaybemoremanageable.Andalsopleasekeepinmindthattheculminating
sections in each chapter are only meant for enticement; in fact, they are actually
meanttoraisejustasmanyquestionsastheyanswer.Thematerialinthelastsections
is not relevant to the core material in subsequent chapters, so doing things a little
outofordershouldcausenoconfusion.However,beawarethatIfindthematerial