Mathematical Analysis Fundamentals

348 Pages·2014·4.45 MB·english

by  Agamirza Bashirov (Auth.)

Mathematical Analysis Fundamentals
Mathematical Analysis
Fundamentals
A. E. Bashirov
Department of Mathematics,
Eastern Mediterranean University, Turkey
and
Institute of Cybernetics, ANAS, Baku, Azerbaijan
E-mail: [email protected]
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Dedicated to Professor Arif Babayev
Preface
Roughly speaking, analysis covers more than half of the whole of mathematics.
It includes the topics following the limit operation and provides a strong basis for
applications of mathematics. Its starting part in the educational process, mathematical
analysis, deals with the issues concentrated around continuity.
Many books have been written on mathematical analysis. The list includes a wide
spectrum of topics from differential and integral calculus for engineers to the highest
mathematical standards such as the books Treatise on Modern Analysis by Dieudonne1
and Principles of Mathematical Analysis by Rudin.2 This book occupies a wide range
in this spectrum. Its narrower audience includes those who are more or less familiar
with differential and integral calculus and would like to get a stronger mathematical
background, but the wider audience covers everyone who wants to get rigorous
fundamentals in analysis. Therefore, it is suggested as a real analysis textbook for
second- or third-year students who have studied differential and integral calculus their
first year. At the same time it may serve as a real analysis textbook for first-year stu-
dents of mathematical departments.
At present, the worldwide number of students is significantly large and is continu-
ously increasing. As a result, the level of an average freshman student is low and is
going to get lower. Currently, the classic method of teaching analysis on the rigorous
level from the beginning may be acceptable in a restricted number of universities. A
large number of universities make a compromise by teaching calculus (i.e., calcu-
lation-based analysis) during freshman year and then set up rigorous mathematical
analysis during sophomore or junior years. This is a solution to the problem, but it
creates an educational problem. It is not easy to change the orientation of sophomore
or junior students from the problems of calculation nature, which are typical for cal-
culus, to rigorous mathematical analysis. This book aims to serve as a transition from
calculus to rigorous analysis.
There have been several books written for this purpose. An important point in
writing such a book is to offer a proper transition rate from elementary calculus to
rigorous analysis. In my opinion, most of the existing books for this purpose either
start from an elementary level and then slightly increase to an intermediate level,
or start from an intermediate or upper intermediate level and quickly progress to an
advanced level. The books of the first kind are not able to cover many useful topics
1 Jean Alexander Ejen Dieudonne (1906–1992), French mathematician. He was one of the active members
of the group of m athematicians developing modern mathematics on the basis of an axiomatic approach
and publishing books under the pseudonym Nicolas Bourbaki.
2 Walter Rudin (1921–2010), U.S. mathematician. His books are very popular in the mathematical
community.
xii Preface
in analysis, and those of the second kind do not provide a sufficiently smooth transi-
tion from calculus to analysis. The goal in this book is a rigorous presentation of the
fundamentals of analysis, starting from an elementary level and smoothly increasing
up to an advanced level.
Familiarity with calculus is not a prerequisite for this book. Rather it is a disad-
vantage with an orientation to the problems of calculation nature. A reorientation to
mathematically rigorous analysis needs effort, especially in starting chapters.
Basic features of this book can be summarized as follows:
• The level of presentation increases, from simple to rigorous throughout the book
as well as within chapters.
• The sequence of abstract reasoning is on the first place. In particular, the principle
of early transcendental functions, which is popular for early creation of valuable
examples, is disregarded.
• Normally, high-level books on analysis do not contain any interpretation with
illustrations, encouraging an analytic way of understanding the subjects. Being
familiar with calculus disregards this principle and, therefore, this book widely
uses graphical interpretations.
• The book invites the reader to think abstractly. Each chapter has a list of exer-
cises, which are mainly designed to clarify the meanings of the definitions and
theorems, to force understanding the proofs, and to call attention to points in the
proofs that might be overlooked.
• One of the aims is to fill the gaps preceding analysis, such as through a clear
understanding of sets, an understanding of proof techniques, and creating a pas-
sage from numbers to systems of numbers. Ordinarily these subjects are included
in appendices. This book intentionally includes these preanalysis subjects as its
integral part.
• The book contains topics for further reading, indicated by some of the sections
being marked by an asterisk. These sections concern relevant topics from func-
tional analysis, measure and integration, differential equations, and some modern
topics such as multiplicative calculus and extension of integrals.
• Brief information about the mathematicians mentioned in the book are given in
the footnotes.
The book is written based on a combination of lecture notes from different courses
on analysis from graduate and undergraduate levels for students familiar with calculus.
One of the difficulties of teaching analysis to students who have already seen
calculus is the transition from solving problems of a computational nature by memo-
rizing the algorithms of solutions, which is typical for calculus, to reading and writ-
ing proofs and understanding how the algorithms of solutions are created. Therefore,
Chapter 1 gently develops these skills in conjunction with a clear understanding of the
concept of a set. Chapter 2 also gently introduces the numbers from the set-theoretic
concepts and extends the subject to cardinality. The relationship between two fields
of numbers, rational and real, is emphasized.
Convergence is an underlying concept of analysis. It is discussed in two chapters.
Chapter 3 deals with convergence on the system of real numbers. Attention is paid
Preface xiii
to the fact that numerical series are just another, but often more convenient, form of
numerical sequences. It is pointed out that the absolutely convergent series behave
similar to finite sums. The distinct features of the system of real numbers are inter-
related to convergence.
A deeper understanding of the concept of convergence comes with metric spaces,
which may or may not have features of the system of real numbers. Study of conver-
gence in metric spaces is important for purposes of generalization and covering many
cases. It is also important to emphasize features of the system of real numbers that can
be overlooked within the frame of real numbers. These topics are discussed in Chapter 4.
The concept of continuity is also discussed in two chapters. Chapter 5 covers
the definition and relationship of continuity and different concepts such as limit,
compactness, and connectedness. Chapter 6 gives a general look at continuity and
discusses the space of continuous functions and its different features as a metric space.
Chapter 7 contains a standard presentation of differentiation with an emphasis on
its relationship to continuity and mean-value theorems. It also contains two sections
for further reading that deal with differential equations and the space of differentiable
functions.
Chapter 8 deals with the concept of bounded variation. Many textbooks cover this
subject by considering only monotone functions. This book goes beyond discussing
discrete and continuous functions of bounded variables and their space.
Integration is also discussed in two chapters. Chapter 9 contains standard explana-
tory material, and Chapter 10 gives an overall view to the integration of functions of
a single variable.
The creation of functions and transcending the capacity of rational functions are
presented in Chapter 11. These functions played a crucial role in the development of
classic mathematical analysis. An interesting feature of this chapter is an introduction
to multiplicative calculus that breaks down the thought about absoluteness of familiar
ordinary calculus.
Finally, Chapter 12 discusses trigonometric series and integrals. The subject is
handled from different kinds of convergence point of views, rather than the calcula-
tion of different trigonometric expansions.
For any questions, comments, or suggestions regarding this book please contact
Agamirza Bashirov at [email protected].
Acknowledgments
I would like to thank my students who taught me how to teach analysis for those
familiar with elementary calculus, all my colleagues from Eastern Mediterranean
University, and all my professors from Baku State University who taught me courses
in different areas of analysis. A special thanks to the late professor Arif Babayev, to
whom this book is dedicated, for his excellent and high-level course in mathematical
analysis. I was one of the lucky students to have taken this complete course during
two academic years in 1971–1972 and 1972–1973.
Thanks to the Elsevier team working on this book, especially Paula Callaghan, Erin
Hill-Parks and Stalin Viswanathan with whom I contacted during the evaluation process.
1
Sets and Proofs
Mathematicsdealswithagreatvarietyofmathematicalconcepts,eachhavingapre-
cisemathematicaldefinition.Unliketheexplanatorymethod,adoptedinmonolingual
dictionaries, mathematical definitions are hierarchical: each of them uses only those
conceptsthataredefinedpreviously.Runningbackalongthishierarchyofmathemat-
icaldefinitions,afterall,onecangetabasicconceptforalltheothers—theconceptof
aset.Sincethereisnothinginmathematicsforegoingtheprimitiveconceptofaset,
oneshouldcarefullyworkwithsetsbyhandlingthemunderasystemofaxiomsthat
excludes an occurrence of paradoxical cases and at the same time preserves a wide
rangeofmanipulations.
Inthischaptersetsandset-theoreticconceptsarediscussed.Althoughadiscussion
of the axioms of set theory lies out of the scope of analysis here, they are briefly
mentionedattheendofthischapter.Anotherfeatureofthischapterisanemphasison
techniquesofproof.Anabilitytoreadandwritemathematicalproofsisveryimportant
tostudyanalysis.Respectively,everyproofgiveninthischapterisaccompaniedwith
adetaileddiscussion.
1.1 Sets, Elements, and Subsets
Asetisaprimitiveconceptofmathematics.Intuitively,itisunderstoodasacollection
ofobjectsthatarecalleditselementsormembers.Sometimesweprefertocallasetas
aclass,system,orfamily.
It is more desirable to symbolize sets by capital letters and their elements by
lowercase letters. In this connection, the different groups of lowercase letters are
usedtodenotedifferentkindsofelements.Intherest,wewillseektousetheletters
a,b,c,... for elements that are fixed for the problem under consideration
(parameters), x,y,z,... for unknowns and for variables, f,g,h,... for functions,
n,m,k,...forintegers, p,q,r,...forelementsofmetricspaces,ε,δ,σ,...forsmall
values,etc.,thoughwedonotmakeanystrictconventionabouttheseusages.
Bothsymbolsa ∈ A and A (cid:3) a meanthata isanelementoftheset A ortheset
A contains a as its element. Similarly, both a ∈/ A and A (cid:4)(cid:3) a mean thata is not an
elementoftheset Aortheset Adoesnotcontainaasitselement.
Asetcanbegivenbylistingallitselementsbetweenbraces.Forexample,theset
{a,b,c}
MathematicalAnalysisFundamentals.http://dx.doi.org/10.1016/B978-0-12-801001-3.00001-9
©2014ElsevierInc.Allrightsreserved.
2 MathematicalAnalysisFundamentals
consistsofthethreeelementsa,b,andc.Oftenweusesetsidentifiedintheform
{a,b,c,...}
ifthereisnoambiguitywiththeelementsmentionedbythethreedots.Asetcanalso
begivenasacollectionofallelementsofacertainsethavingacertainproperty.For
example,theset
{a ∈ A:ahastheproperty P}
consistsofallelementsoftheset Athathavetheproperty P.Thesymbol
{a :ahastheproperty P}
maybeusedfortheprecedingsetifitisclearwhatisA.Noticethatthissymbolwithout
anyexistingset Amaycauseacontradiction.Forexample,
{a :aisaset}
doesnotexistasaset.
Everysetiscompletelydeterminedbyknowledgeofallitselementsandbynothing
else.Thissimpleremarkhasafewusefulconsequences.First,itdefinesacriterionfor
equalityofsets:twosets Aand B areequaliftheyconsistofthesameelements;this
isindicatedas A= B.Otherwise,wewrite A(cid:4)= B.
Second,itimpliesthatanyrearrangementaswellasanyrepetitionoftheelements
do not change the set. Consequently, when we identify a set by listing its elements,
we usually list only its distinct elements disregarding their order. For example, the
symbols{a,b}and{b,a}representthesamesetconsistingoftwodistinctelementsa
andb(symbolically,a (cid:4)=b).Ifa =b(aandbareequalelements),thenwewrite{a}
or{b}insteadof{a,b}and{b,a}.
Finally,itnotifiesthatbeforeformingasetasacollectionofobjects,atfirstthese
objectsmustbeavailable.Consequently,itfollowsthatthereisnosetcontainingitself
as an element of itself. For example, the expression A = {A,a,b,c,...} does not
define Aasaset.
Asetcontainingonlyoneelementiscalledasingleton.Thesymbol{a}expresses
thesingleton,containingaasitselement.Theemptyset∅isasetthatdoesnotcontain
any element. One must distinguish ∅ and {∅}, the first of them being the empty set
andthesecondasingleton.
If Aand B aretwosetssothateveryelementof Aisanelementof B,thenwesay
that Aisasubsetof B or B isasupsetof A;thisisindicatedas
A⊆ B or B ⊇ A.
Clearly,foreveryset A,itistruethat∅ ⊆ A and A ⊆ A.Wesaythat A isaproper
subsetof B or B isapropersupsetof Aif A⊆ B and A(cid:4)= B;thisisindicatedas
A⊂ B or B ⊃ A.
SetsandProofs 3
•
•
(a) (b) (c) (d)
A
(e) (f) (g) (h)
Figure1.1 Venndiagrams.
If Aisnotasubsetof B,thenwewrite
A(cid:4)⊆ B or B (cid:4)⊇ A.
Onemustcorrectlyusethesymbols∈and⊆(respectively,(cid:3)and⊇)anddistinguish
anelementfromasubset.Forthis,thefollowingguidesareuseful:
element∈set, subset⊆set.
In fact, a ∈ A implies {a} ⊆ A and vice versa. The symbols ∈,∈/,⊆, and (cid:4)⊆ are
graphicallydemonstratedbyVenn3 diagramsinFigure1.1(a)–(d).
1.2 Operations on Sets
Byuseofoperationsonsets,weformnewsets.Let Aand Bbetwosets.Theunionof
AandBisthesetconsistingofallelementsof AandB;itisdenotedby A∪B,thatis,
A∪B ={a :a ∈ Aora ∈ B}.
Theintersectionof Aand B isthesetconsistingofallcommonelementsof Aand B;
thesymbol A∩B isusedforthisset,thatis,
A∩B ={a :a ∈ Aanda ∈ B}.
3JohnVenn(1834–1923),Englishmathematicianandlogician.Heusedso-calledVenndiagramstopopu-
larizesymboliclogic.