Introduction to Analysis in One Variable

289 Pages·2020·1.105 MB·English

by  Michael E. Taylor

Introduction to Analysis in One Variable
Michael Taylor
Math. Dept., UNC
E-mail address: [email protected]
2010 Mathematics Subject Classi(cid:12)cation. 26A03, 26A06, 26A09, 26A42
Key words and phrases. real numbers, complex numbers, irrational
numbers, Euclidean space, metric spaces, compact spaces, Cauchy
sequences, continuous function, power series, derivative, mean value
theorem, Riemann integral, fundamental theorem of calculus, arclength,
exponential function, logarithm, trigonometric functions, Euler’s formula,
Weierstrass approximation theorem, Fourier series, Newton’s method

Contents
Preface ix
Some basic notation xiii
Chapter 1. Numbers 1
x1.1. Peano arithmetic 2
x1.2. The integers 9
x1.3. Primefactorizationandthefundamentaltheoremofarithmetic 14
x1.4. The rational numbers 16
x1.5. Sequences 21
x1.6. The real numbers 29
x1.7. Irrational numbers 41
x1.8. Cardinal numbers 44
x1.9. Metric properties of R 51
x1.10. Complex numbers 56
Chapter 2. Spaces 65
x2.1. Euclidean spaces 66
x2.2. Metric spaces 74
x2.3. Compactness 80
x2.4. The Baire category theorem 85
Chapter 3. Functions 87
x3.1. Continuous functions 88
x3.2. Sequences and series of functions 97
vii
viii Contents
x3.3. Power series 102
x3.4. Spaces of functions 108
x3.5. Absolutely convergent series 112
Chapter 4. Calculus 117
x4.1. The derivative 119
x4.2. The integral 129
x4.3. Power series 148
x4.4. Curves and arc length 158
x4.5. The exponential and trigonometric functions 172
x4.6. Unbounded integrable functions 191
Chapter 5. Further Topics in Analysis 199
x5.1. Convolutions and bump functions 200
x5.2. The Weierstrass approximation theorem 205
x5.3. The Stone-Weierstrass theorem 208
x5.4. Fourier series 212
x5.5. Newton’s method 237
x5.6. Inner product spaces 243
Appendix A. Complementary results 247
xA.1. The fundamental theorem of algebra 247
xA.2. More on the power series of (1(cid:0)x)b 249
xA.3. (cid:25)2 is irrational 251
xA.4. Archimedes’ approximation to (cid:25) 253
xA.5. Computing (cid:25) using arctangents 257
xA.6. Power series for tanx 261
xA.7. Abel’s power series theorem 264
xA.8. Continuous but nowhere-differentiable functions 268
Bibliography 273
Index 275
Preface
This is a text for students who have had a three course calculus sequence,
and who are ready for a course that explores the logical structure of this
areaofmathematics, whichformsthebackboneofanalysis. Thisisintended
for a one semester course. An accompanying text, Introduction to Analysis
in Several Variables [13], can be used in the second semester of a one year
sequence.
The main goal of Chapter 1 is to develop the real number system. We
start with a treatment of the \natural numbers" N, obtaining its structure
fromashortlistofaxioms, theprimaryonebeingtheprincipleofinduction.
Then we construct the set Z of all integers, which has a richer algebraic
structure, and proceed to construct the set Q of rational numbers, which
are quotients of integers (with a nonzero denominator). After discussing
in(cid:12)nite sequences of rational numbers, including the notions of convergent
sequences and Cauchy sequences, we construct the set R of real numbers,
as ideal limits of Cauchy sequences of rational numbers. At the heart of
this chapter is the proof that R is complete, i.e., Cauchy sequences of real
numbers always converge to a limit in R. This provides the key to studying
other metric properties of R, such as the compactness of (nonempty) closed,
bounded subsets. We end Chapter 1 with a section on the set C of complex
numbers. Many introductions to analysis shy away from the use of complex
numbers. My feeling is that this forecloses the study of way too many
beautiful results that can be appreciated at this level. This is not a course
incomplexanalysis. Thatisforanothercourse, andwithanothertext(such
as [14]). However, I think the use of complex numbers in this text serves
both to simplify the treatment of a number of key concepts, and to extend
their scope in natural and useful ways.
ix
x Preface
In fact, the structure of analysis is revealed more clearly by moving
beyond R and C, and we undertake this in Chapter 2. We start with a
treatment of n-dimensional Euclidean space, Rn. There is a notion of Eu-
clideandistancebetweentwopointsinRn, leadingtonotionsofconvergence
and of Cauchy sequences. The spaces Rn are all complete, and again closed
bounded sets are compact. Going through this sets one up to appreciate a
further generalization, the notion of a metric space, introduced in x2.2. This
is followed by x2.3, exploring the notion of compactness in a metric space
setting.
Chapter 3 deals with functions. It starts in a general setting, of func-
tions from one metric space to another. We then treat in(cid:12)nite sequences
of functions, and study the notion of convergence, particularly of uniform
convergence of a sequence of functions. We move on to in(cid:12)nite series. In
such a case, we take the target space to be Rn, so we can add functions.
Section 3.3 treats power series. Here, we study series of the form
∑1
(0.0.1) a (z(cid:0)z )k;
k 0
k=0
with a 2 C and z running over a disk in C. For results obtained in this
k
section, regardingtheradiusofconvergenceR andthecontinuityofthesum
onD (z ) = fz 2 C : jz(cid:0)z j < Rg,thereisnoextradifficultyinallowinga
R 0 0 k
and z to be complex, rather than insisting they be real, and the extra level
of generality will pay big dividends in Chapter 4. One section in Chapter 3
is devoted to spaces of functions, illustrating the utility of studying spaces
beyond the case of Rn.
Chapter 4 gets to the heart of the matter, a rigorous development of dif-
ferential and integral calculus. We de(cid:12)ne the derivative in x4.1, and prove
the Mean Value Theorem, making essential use of compactness of a closed,
bounded interval and its consequences, established in earlier chapters. This
result has many important consequences, such as the Inverse Function The-
orem, and especially the Fundamental Theorem of Calculus, established in
x4.2, after the Riemann integral is introduced. In x4.3, we return to power
series, this time of the form
∑1
(0.0.2) a (t(cid:0)t )k:
k 0
k=0
We require t and t to be in R, but still allow a 2 C. Results on radius
0 k
of convergence R and continuity of the sum f(t) on (t (cid:0)R;t +R) follow
0 0
from material in Chapter 3. The essential new result in x4.3 is that one can
′
obtain the derivative f (t) by differentiating the power series for f(t) term
by term. In x4.4 we consider curves in Rn, and obtain a formula for arc
length for a smooth curve. We show that a smooth curve with nonvanishing
Preface xi
velocity can be parametrized by arc length. When this is applied to the unit
circle in R2 centered at the origin, one is looking at the standard de(cid:12)nition
of the trigonometric functions,
(0.0.3) C(t) = (cost;sint):
We provide a demonstration that
(0.0.4) C′(t) = ((cid:0)sint;cost)
that is much shorter than what is usually presented in calculus texts. In
x4.5 we move on to exponential functions. We derive the power series for
the function et, introduced to solve the differential equation dx=dt = x. We
then observe that with no extra work we get an analogous power series for
eat, with derivative aeat, and that this works for complex a as well as for
real a. It is a short step to realize that eit is a unit speed curve tracing out
the unit circle in C (cid:25) R2, so comparison with (0.0.3) gives Euler’s formula
(0.0.5) eit = cost+isint:
That the derivative of eit is ieit provides a second proof of (0.0.4). Thus
we have a uni(cid:12)ed treatment of the exponential and trigonometric functions,
carried out further in x4.5, with details developed in numerous exercises.
Section4.6extendsthescopeoftheRiemannintegraltoaclassofunbounded
functions.
Chapter 5 treats further topics in analysis. The topics center around
approximating functions, via various in(cid:12)nite sequences or series. Topics
include polynomial approximation of continuous functions, Fourier series,
and Newton’s method for approximating the inverse of a given function.
We end with a collection of appendices, covering various results related
to material in Chapters 4{5. The (cid:12)rst one gives a proof of the fundamental
theorem of algebra, that every nonconstant polynomial has a complex root.
The second explores the power series of (1(cid:0)x)b, in more detail then dome
in x4.3, of use in x5.2. There follow three appendices on the nature of (cid:25) and
its numerical evaluation, an appendix on the power series of tanx, and one
on a theorem of Abel on in(cid:12)nite series, and related results. We also study
continuous functions on R that are nowhere differentiable.
Our approach to the foundations of analysis, outlined above, has some
distinctive features, which we point out here.
1) Approach to numbers. We do not take an axiomatic approach to the
presentation of the real numbers. Rather than hypothesizing that R has
speci(cid:12)edalgebraicandmetricproperties,webuildRfrommorebasicobjects
xii Preface
(natural numbers, integers, rational numbers) and produce results on its
algebraic and metric properties as propositions, rather than as axioms.
In addition, we do not shy away from the use of complex numbers. The
simpli(cid:12)cationsthisuseaffordsrangefromamusing(constructionofaregular
pentagon) to profound (Euler’s identity, computing the Dirichlet kernel in
Fourierseries),andsuchusesofcomplexnumberscanbereadilyappreciated
by a student at the level of this sort of analysis course.
2) Spaces and geometrical concepts. We emphasize the use of geometrical
properties of n-dimensional Euclidean space, Rn, as an important extension
of metric properties of the line and the plane. Going further, we introduce
the notion of metric spaces early on, as a natural extension of the class of
Euclidean spaces. For one interested in functions of one real variable, it is
very useful to encounter such functions taking values in Rn (i.e., curves),
and to encounter spaces of functions of one variable (a signi(cid:12)cant class of
metric spaces).
One implementation of this approach involves de(cid:12)ning the exponential
function for complex arguments and making a direct geometrical study of
eit, for real t. This allows for a self-contained treatment of the trigonomet-
ric functions, not relying on how this topic might have been covered in a
previous course, and in particular for a derivation of the Euler identity that
is very much different from what one typically sees.
Wefollowthisintroductionwitharecordofsomestandardnotationthat
will be used throughout this text.
Acknowledgment
During the preparation of this book, I have been supported by a number of
NSF grants, most recently DMS-1500817.