An introduction to analysis

373 Pages·2017·1.635 MB·English

by  Robert C. Guning

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[chapter] [chapter]
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AN INTRODUCTION TO ANALYSIS
ROBERT C. GUNING
(cid:13)cRobert C. Gunning
ii
Introduction
Thecoursenotesonwhichthisbookisbasedservedasthetextfortheaccel-
eratedhonorsanalysiscoursethatIhavebeenteachingatPrincetonUniversity.
ThetraditionalhonorsmathematicsprogramatPrincetonconsistsofthreeone-
semester courses, covering calculus in one variable in the first semester, linear
algebra in the second semester, and calculus in several variables in the third
semester,andwasusuallytakenbyundergraduatesintheirfirstthreesemesters.
Inrecentyearsundergraduateshaveenteredtheuniversitywithmoreexperience
in rigorous and abstract mathematics, and have been particularly eager to pass
on to more advanced mathematics courses as quickly as possible. Consequently
as an experiment we tried condensing the contents of the three semesters into a
one-year course, with a fair bit of additional material as well, called the accel-
erated honors analysis course. About a third of the students taking the honors
sequence have chosen the accelerated version, and generally they have done
quitewellindeed. Thecoursebeginsratherabstractlywithabackgroundinset
theory and algebra, treats differentiation and integration in several dimensions
from the beginning, treats some standard material on linear algebra of particu-
lar interest in analysis, and ends with differential forms and Stokes’s theorems
in n dimensions, with the classical cases as examples. The course covers a good
dealofmaterialandproceedsquiterapidly. Itmeetstwiceweekly,for90minute
sessions, but in addition there are tutorial sessions three times a week in the
evening, run by student tutors to help the students read and understand the
somewhat densely written course notes and to provide guidance in aproaching
the problems. Providing these extra sessions is possibly the only way to cover
the material at a not too unreasonable pace. Those students who feel that the
lectures are rather rushed do turn up quite regularly for the tutorial sessions.
During the 2007-2008 academic year Lillian Pierce and I cotaught the third
term of the regular honors course, while she was a graduate student here, and
we reorganized the presentation of the material and the problem assignments;
her suggestions were substantial and she played a significant part in working
through the revisions that year. The goal was to continue the rather abstract
treatmentoftheunderlyingmathematicaltopicsfollowingMichaelSpivak’snow
classical treatment, but with somewhat more emphasis on differentiation and
somewhat less emphasis on the abstract treatment of differential forms. The
problems were divided into two categories: a first group of problems testing
a basic understanding of the essential topics discussed and their applications,
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iv INTRODUCTION
problems that almost all serious students should be able to solve without too
much difficulty, and a second group of problems covering more theoretical as-
pectsandtoughercalculations,challengingthestudentsbutstillnotparticularly
difficult. Thetemptationtoincludeathirdcategoryofoptionalverychallenging
problems, to introduce interested students to a wider range of other theoretical
and practical aspects, was frustrated by a resounding lack of interest on the
partofthestudents. Whentheacceleratedcoursewassetupinthe2010-2011
academic year I used the notes we had developed for the third term of the reg-
ular course sequence and added additional material at the beginning, including
more linear algebra.
Ishouldliketoexpressheremysincerethankstothestudentswhocompiled
the lecture notes that were the basis for the first version of the course notes:
Robert Haraway, Adam Hesterberg, Jay Holt and Alex Schiller; to the gradu-
ate students who have cotaught the course and have done the major share of
grading the assignments; and to the students who have taken the course these
pastfewyears, foranumberofcorrectionsandsuggestionsforgreaterclarity. I
would particularly like to thank my colleagues Lillian Pierce, for her many sug-
gestions and very great help in reorganizing the course, and Tadahiro Oh, for a
very thorough review of the notes and a great many very valuable suggestions
andcorrections. IoweaparticulardebtofgratitudetoKennethTamforavery
closereadingofthelastversionsofthenotes. Hepointedoutandhelpedcorrect
some errors that had slipped into previous revisions; he was an excellent critic
on matters of consistency and clarity; and he suggested some very significant
improvements and additions. The remaining errors and confusion are my own
responsibility though.
Robert C. Gunning
Fine Hall, Princeton University, Summer 2017
Contents
Introduction iii
1 Algebraic Fundamentals 1
1.1 Sets and Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Groups, Rings and Fields . . . . . . . . . . . . . . . . . . . . . . 14
1.3 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2 Topological Fundamentals 61
2.1 Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.2 Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.3 Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 86
2.4 Compact Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3 Mappings 109
3.1 Continuous Mappings . . . . . . . . . . . . . . . . . . . . . . . . 109
3.2 Differentiable Mappings . . . . . . . . . . . . . . . . . . . . . . . 129
3.3 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
4 Linear Mappings 185
4.1 Endomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
4.2 Inner Product Spaces. . . . . . . . . . . . . . . . . . . . . . . . . 206
5 Geometry of Mappings 225
5.1 The Inverse Mapping Theorem . . . . . . . . . . . . . . . . . . . 225
5.2 The Implicit Function Theorem . . . . . . . . . . . . . . . . . . . 240
5.3 The Rank Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . 249
6 Integration 259
6.1 Riemann Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
6.2 Caculation of Integrals . . . . . . . . . . . . . . . . . . . . . . . . 278
7 Differential Forms 297
7.1 Line Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
7.2 Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . 311
7.3 Integrals of Differential Forms . . . . . . . . . . . . . . . . . . . . 329
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vi CONTENTS
Chapter 1
Algebraic Fundamentals
1.1 Sets and Numbers
Perhapsthebasicconceptinmathematicsisthatofaset,anunambiguously
determinedcollectionofmathematicalentities. Asetcanbespecifiedbylisting
itscontents,asforinstancethesetA={w,x,y,z},orbydescribingitscontents
inanyotherwaysolongasthemembersofthesetarefullydetermined. Among
all sets the strangest is no doubt the empty set, traditionally denoted by ∅,
the set that contains nothing at all, so the set ∅ = { }; this set should be
approachedwithsomecaution,sinceitcanreadilysneakupunexpectedlyasan
exceptionorcounterexampletosomemathematicalstatementifthatstatement
isnotcarefullyphrased. Thereisagenerallyacceptedstandardnotationdealing
with sets, and students should learn it and use it freely. That a belongs to or
is a member of the set A is indicated by writing a∈A; that b does not belong
to the set A is indicated by writing b (cid:54)∈ A. A set A is a subset of a set B if
whenever a ∈ A then also a ∈ B, and the condition that A is a subset of B is
indicated by writing A ⊂ B or B ⊃ A. There is some variation in the usage
though. With the definition adopted here, A⊂B includes the case that A=B
as well as the case that there are some b ∈ B for which b (cid:54)∈ A; in the latter
casetheinclusionA⊂B issaidtobeaproper inclusionanditisdenotedby
A(cid:36)B or B (cid:37)A . In particular ∅⊂A for any set A; for since there is nothing
in the empty set ∅ the condition that anything contained in ∅ is also contained
in A holds vacuously. Sets A and B are said to be equal, denoted by A = B
or B = A, if they contain exactly the same elements, or equivalently if both
A⊂B and B ⊂A.
The intersection of sets A and B, denoted by A ∩ B, consists of those
elementsthatbelongtobothAandB,andtheunionofsetsAandB,denoted
by A ∪ B, consists of those elements that belong to either A or B or both.
More generally, for any collection of sets {A }, the intersection and union of
α
1
2 CHAPTER 1. ALGEBRAIC FUNDAMENTALS
Figure1.1: VenndiagramsillustratingthesetsA∩B, A∪B, A∆B respectively.
this collection of sets are defined by
(cid:92) (cid:110) (cid:12) (cid:111)
(1.1) A = a(cid:12)a∈A for all A ,
α (cid:12) α α
α
(cid:91) (cid:110) (cid:12) (cid:111)
A = a(cid:12)a∈A for some A .
α (cid:12) α α
α
Two sets A and B are disjoint if A ∩ B = ∅. The intersection and union
operations on sets are related by
(1.2) A∩(B∪C)=(A∩B)∪(A∩C),
A∪(B∩C)=(A∪B)∩(A∪C).
ThedifferenceoftwosetsAandB,denotedbyA∼B,consistsofthosea∈A
such that a (cid:54)∈ B, whether B is a subset of A or not; for clarity ∅ ∼ ∅ = ∅, a
result that might not be altogether clear from the definition of the difference.
Obviously the order in which the two sets are listed is critical in this case; but
there is also the symmetric difference between these two sets, defined by
(1.3) A∆B =B∆A=(A∼B)∪(B ∼A)=(A∪B)∼(A∩B).
The symmetric difference, as well as the intersection and union, can be illus-
trated by Venn diagrams, as in the accompanying Figure 1.1. It is clear from
the definitions that
(1.4) A∼(B∪C)=(A∼B)∩(A∼C),
A∼(B∩C)=(A∼B)∪(A∼C).
If A,B,C,... are all viewed as subsets of a set E, and that is understood in the
discussionofthesesubsets,thenadifferencesuchasE ∼Aoftenisdenotedjust
by∼AandiscalledthecomplementofthesubsetA;withthisunderstanding
(1.4) takes the form
(1.5) ∼(A∪B)=(∼A)∩(∼B),
∼(A∩B)=(∼A)∪(∼B).
It is worth writing out the proofs of (1.2) and (1.4) in detail if these equations
do not seem evident.
A mapping f : A −→ B from a set A to a set B associates to each a ∈ A
its image f(a)∈B. The set A often is called the domain of the mapping and