Davidson K.R., Donsig A.P. Real Analysis with Real Applications

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PREFACE

This book provides an introduction both to real analysis and to a range of important

applications that require this material. More than half the book is a series of es-

sentially independent chapters covering topics from Fourier series and polynomial

approximation to discrete dynamical systems and convex optimization. Studying

these applications can, we believe, both improve understanding of real analysis and

prepare for more intensive work in each topic. There is enough material to allow a

choice of applications and to support courses at a variety of levels.

The ﬁrst part of the book covers the basic machinery of real analysis, focusing

on that part needed to treat the applications. This material is organized to allow a

streamlined approach that gets to the applications quickly, or a more wide-ranging

introduction. To this end, certain sections have been marked as enrichment topics

or as advanced topics to suggest that they might be omitted. It is our intent that

the instructor will choose topics judiciously in order to leave sufﬁcient time for

material in the second part of the book.

A quick look at the table of contents should convince the reader that applica-

tions are more than a passing fancy in this book. Material has been chosen from

both classical and modern topics of interest in applied mathematics and related

ﬁelds. Our goal is to discuss the theoretical underpinnings of these applied areas

concentrating on the role of fundamental principles of analysis. This is not a meth-

ods course, although some familiarity with the computational or methods-oriented

aspects of these topics may help the student appreciate how the topics are devel-

oped. In each application, we have attempted to get to a number of substantial

results and to show how these results depend on the fundamental ideas of real anal-

ysis. In particular, the notions of limit and approximation are two sides of the same

coin, and this interplay is central to the whole book.

We emphasize the role of normed vector spaces in analysis, as they provide a

natural framework for most of the applications. This begins early with a separate

treatment of R

. Normed vector spaces are introduced to study completeness and

limits of functions. There is a separate chapter on metric spaces that we use as

an opportunity to put in a few more sophisticated ideas. This format allows its

omission, if need be.

The basic ideas of calculus are covered carefully, as this level of rigour is not

generally possible in a ﬁrst calculus course. One could spend a whole semester

doing this material, which forms the basis of many standard analysis courses today.

When we have taught a course from these notes, however, we have often chosen to

omit topics such as the basics of differentiation and integration on the grounds that

these topics have been covered adequately for many students. The goal of getting

further into the applications chapters may make it worth cutting here.

We have treated only tangentially some topics commonly covered in real anal-

ysis texts, such as multivariate calculus or a brief development of the Lebesgue

xii Preface

integral. To cover this material in an accessible way would have left no time, even

in a one-year course, for the real goal of the book. Nevertheless, we deal through-

out with functions on domains in R

, and we do manage to deal with issues of

higher dimensions without differentiability. For example, the chapter on convexity

and optimization yields some deep results on “nonsmooth” analysis that contain

the standard differentiable results such as Lagrange multipliers. This is possible

because the subject is based on directional derivatives, an essentially one-variable

idea. Ideas from multivariate calculus appear once or twice in the advanced sec-

tions, such as the use of Green’s Theorem in the section on the isoperimetric in-

equality.

Not covering measure theory was another conscious decision to keep the ma-

terial accessible and to keep the size of the book under control. True, we do make

use of the L

norm and do mention the L

spaces because these are important

ideas. We feel, however, that the basics of Fourier series, approximation theory,

and even wavelets can be developed while keeping measure theory to a minimum.

Of course, this does not mean we think that the subject is unimportant. Rather we

wished to aim the book at an undergraduate audience. To deal partially with some

of the issues that arise here, we have included a section on metric space completion.

This allows a treatment of L

spaces as complete spaces of bona ﬁde functions, by

means of the Daniell integral. This is certainly an enrichment topic, which can be

used to motivate the need for measure theory and to satisfy curious students.

This book began in 1984 when the ﬁrst author wrote a short set of course notes

(120 pages) for a real analysis class at the University of Waterloo designed for

students who came primarily from applied math and computer science. The idea

was to get to the basic results of analysis quickly and then illustrate their role in a

variety of applications. At that time, the applications were limited to polynomial

approximation, Newton’s method, differential equations, and Fourier series.

A plan evolved to expand these notes into a textbook suitable for one semester

or a year-long course. We expanded both the theoretical section and the choice of

applications in order to make the text more ﬂexible. As a consequence, the text

is not uniformly difﬁcult. The material is arranged by topic, and generally each

chapter gets more difﬁcult as one progresses through it. The instructor can choose

to omit some more difﬁculttopics in the chapters on abstract analysis if they will not

be needed later. We provide a ﬂow chart indicating the topics in abstract analysis

required for each part of the applications chapters. For example, the chapter on

limits of functions begins with the basic notion of uniform convergence and the

fundamental result that the uniform limit of continuous functions is continuous. It

ends with much more difﬁcult material, such as the Arzela–Ascoli Theorem. Even

if one plans to do the chapter on differential equations, it is possible to stop before

the last section on Peano’s Theorem, where the Arzela–Ascoli Theorem is needed.

So both topics can be conveniently omitted. Although one cannot proceed linearly

through the text, we hope there is some compensation in demonstrating that, even

at a high level, there is a continued interplay between theory and application.

The background assumed for using this text is decent courses in both calculus

and linear algebra. What we expect is outlined in the background chapter. A student

should have a reasonable working knowledge of differential and integral calculus.

Preface xiii

Multivariable calculus is an asset because of the increased level of sophistication

and the incorporation of linear algebra; it is not essential. We certainly expect that

the student is used to working with exponentials, logarithms, and trigonometric

functions. Linear algebra is needed because we treat R

, C(X), and L

(−π, π) as

vector spaces. We develop the notion of norms on vector spaces as an important

tool for measuring convergence. As such, the reader should be comfortable with

the notion of a basis in ﬁnite-dimensional spaces. Familiarity with linear transfor-

mations is also sometimes useful. A course that introduces the student to proofs

would also be an asset. Although we have attempted to address this in the back-

ground chapter (Chapter 1), we have no illusions that this text would be easy for a

student having no prior experience with writing proofs.

While this background is in principle enough for the whole book, sections

marked with a • require additional mathematical maturity or are not central to

the main development, and sections marked with a ? are more difﬁcult yet. By

and large, the various applications are independent of each other. However, there

are references to material in other chapters. For example, in the wavelets chapter

(Chapter 15), it seems essential to make comparisons with the classical approxima-

tion results for Fourier series and for polynomials.

It is also possible to use an application chapter on its own for a student semi-

nar or other topics course. We have included several modern topics of interest in

addition to the classical subjects of applied mathematics. The chapter on discrete

dynamical systems (Chapter 11) introduces the notions of chaos and fractals and

develops a number of examples. The chapter on wavelets (Chapter 15) illustrates

the ideas with the Haar wavelet. It continues with a construction of wavelets of

compact support, and gives a complete treatment of a somewhat easier continuous

wavelet. In the ﬁnal chapter (Chapter 16), we study convex optimization and con-

vex programming. Both of these latter chapters require more linear algebra than

the others.

We would like to thank various people who worked with early versions of this

book for their helpful comments; in particular, Robert Andr

e, John Baker, Brian

Forrest, John Holbrook, David Seigel, and Frank Zorzitto. We also thank various

people who offered us assistance in various ways, including Jon Borwein, Stephen

Krantz, Justin Peters, and Ed Vrscay. We also thank our student Masoud Kamgar-

pour for working through parts of the book. We would particularly like to thank the

students in various classes, at the University of Waterloo and at the University of

Nebraska, where early versions of the text were used.

We welcome comments on this book.

Pure Mathematics Department, University of Waterloo, Waterloo, ON N2L 3G1

Canada

krdavids@math.uwaterloo.ca

Mathematics and Statistics Department, University of Nebraska–Lincoln, Lincoln,

NE 68588-0323, United States

adonsig@math.unl.edu

DEPENDENCE ON EARLIER SECTIONS

The description of dependence on earlier material will be simpliﬁed by assuming

a core of material. Some parts of this may be reasonably omitted by relying on

the student’s background. For example, consider omitting Background, Series, and

Differentiation and Integration.

Core Material

1. Background, 1.1–1.4

2. The Real Numbers, 2.1–2.7

3. Series, 3.1–3.2

4. Topology of R

, 4.1–4.4

5. Functions, 5.1–5.6

6. Differentiation and Integration 6.1–6.4

7. Normed Spaces, 7.1–7.4

8. Limits of Functions, 8.1–8.4

Enrichment Topics. Special topics in Part A, marked with a bullet • for en-

richment or a star ? for advanced, generally may be done with the core material (to

that point) with a few exceptions, which are noted below.

Enrichment Topic Section Dependence

• Equivalence relations 1.5

• Cardinality 2.8 1.5

• e 3.3

• Absolute and conditional convergence 3.4

• Monotone functions 5.7

• Stirling’s formula 6.5

? Measure zero 6.6

• Finite-dimensional normed spaces 7.6

? The L

norms 7.7

• Power series 8.5 3.4

? Arzela–Ascoli theorem 8.6

• Topology in metric spaces 9.2–9.4

? Metric completion 9.5

? The L

spaces 9.6 1.5, 7.7, 9.5

xiv

Dependence on Earlier Sections xv

Applied topics assume the core material and the standard sections preceding in

their own chapter.

Applied Topic Sections Background Required

Polynomial approximation 10.1–10.4 8.5

• Best approximation 10.5–10.6 7.6

? Chebychev expansions 10.7 7.6, 10.6

? Splines 10.8–10.9 linear algebra

? Stone–Weierstrass theorem 10.10 9.1–9.3

Discrete dynamical systems

Contraction principle 11.1

Newton’s method 11.2 10.1,11.1

• Chaos 11.3–11.5

? Topological conjugacy 11.6 5.7

? Fractals 11.7 9.1, 9.2, only 11.1

ODEs 12.1–12.5 11.1

• Linear DEs 12.6

• Perturbations of DEs 12.7

? Peano’s theorem 12.8 8.6

Fourier series and physics 13.1–13.7

• Vibrating String 13.8–13.9

• Complex Exponential 13.10 8.5

Fourier series and approximation 14.1–14.6 7.6, 13.3, 13.5, 13.6

? Best approximation 14.7–14.9 10.1–10.5, 10.7

Wavelets 15.1–15.4 7.6, 10.4

• Daubechies wavelet 15.5–15.7 linear algebra

• Franklin wavelet 15.8–15.9 7.6, 10.8, 13.10, linear algebra

Convexity and optimization 16.1–16.5 5.7, linear algebra

• Constrained optimization 16.6–16.9 multivariable calculus

POSSIBLE COURSE OUTLINES

Core Material (Abridged)

The Real Numbers, 2.1–2.7

Series, 3.1–3.2

Topology of R

, 4.1–4.4

Functions, 5.1–5.6

Normed Spaces, 7.1–7.4

Limits of Functions, 8.1–8.4

A One-Semester Applied Course

Core Material (abridged)

Polynomial Approximation, 10.1–10.5

Contraction Maps and Newton’s Method, 11.1, 11.2

Differential Equations, 12.1–12.4

Fourier Series, 13.1–13.7, 14.1, 14.2

A One-Semester Course with Approximation

Core Material (abridged) Plus 3.4, 7.6, 8.5

Polynomial Approximation, 10.1–10.6, 10.8

Fourier Series and Approximation, 13.3, 13.5–6, Chapter 14

Wavelets, 15.1–15.4, if time permits

A One-Semester Course with Dynamics

Core Material (abridged) Plus 3.4, 9.1

Dynamical Systems, Chapter 11

Differential Equations, Chapter 12

Fourier Series and Physics, Chapter 13

A One-Semester Course with Optimization

Core Material (abridged) Plus 9.1, 8.5, 8.6

Approximation by Polynomials, 10.1–10.6

Convexity and Optimization, Chapter 16

Differential Equations, Chapter 12

A One-Semester Standard Analysis Course

Reals, Series, Topology, Functions, Chapters 2–5

Differentiation and Integration, 6.1–6.4

Normed Spaces, 7.1, 7.2

Limits of Functions, Chapter 8

Metric Spaces, 9.1–9.4

Polynomial Approximation, 10.1–10.3

One application topic, if time permits

xvi

Possible Course Outlines xvii

A Sample One-Year Course

Reals, Series, Topology, Functions, Chapters 2–5

Differentiation and Integration, 6.1–6.5

Normed Spaces, 7.1–7.6

Limits of Functions, Chapter 8

Metric Spaces, 9.1–9.4

Polynomial Approximation, 10.1–10.7

Discrete Dynamical Systems, 11.1–11.5

Fourier Series and Physics, 13.1–13.7

Fourier Series and Approximation, 14.1–14.8

Convexity and Optimization, Chapter 16, if time permits

CHAPTER 1

Background

This chapter covers the language of mathematics and the basic objects, such as

functions, that underlie everything that we do later. It also summarizes what the

reader is expected to know from calculus and linear algebra, the material that real

analysis builds upon. The most important prerequisite is that amorphous attribute

“mathematical maturity,” which means being comfortable with abstraction, being

able to connect abstract statements with concrete examples, and having experience

in reading and writing proofs. Working through this chapter will force you to prac-

tice these skills and will help prepare you for the rest of the book.

1.1. The Language of Mathematics

The language of mathematics has to be precise, because mathematical state-

ments must be interpreted with as little ambiguity as possible. Indeed, the rigour

in mathematics is much greater than in law. There should be no doubts, reason-

able or otherwise, when a theorem is proved. It is either completely correct, or it

is wrong. Consequently, mathematicians have adopted a very precise language so

that statements may not be misconstrued.

In complicated situations, it is easy to fool yourself. By being very precise and

formal now, we can build up a set of tools that will help prevent mistakes later.

The history of mathematics is full of stories in which mathematicians have fooled

themselves with incorrect proofs. Clarity in mathematical language, like clarity in

all other kinds of writing, is essential to communicating your ideas.

We begin with a brief discussion of the logical usage of certain innocuous

words if, then, only if, and, or and not. Let A, B, C represent statements that may

or may not be true in a speciﬁc instance. For example, consider the statements

A. It is raining.

B. The sidewalk is wet.

The statement “If A, then B” means that whenever A is true, it follows that B

must also be true. We also formulate this as “A implies B.” This statement does

not claim either that the sidewalk is wet or that it is not. It tells you that if you

2 Background

look outside and see that it is raining, then without looking at the sidewalk, you

will know that the sidewalk is wet as a result. As in the English language, “if A,

then B” is a conditional statement meaning that only when the hypothesis A is

veriﬁed can you deduce that B is valid. One also writes “Suppose A. Then B”

with essentially the same meaning.

On the other hand, A implies B is quite different from B implies A. For

example, the sidewalk may be wet because

C. The lawn sprinkler is on.

The statement “if B, then A” is known as the converse of “if A, then B.” This

amounts to reversing the direction of the implication. As you can see from this

example, one may be true but not the other.

We can also say “A if B” to mean “if B, then A.” The statement “B only

if A” means that in order that B be true, it is necessary that A be true. A bit of

thought reveals that this is yet another reformulation of “if A, then B.” For reasons

of clarity, these two expressions are rarely used alone and are generally restricted

to the combined statement “A if and only if B.” Parsing this sentence, we arrive at

two statements “A if B” and “A only if B.” The former means “B implies A” and

the latter means “A implies B.” Together they mean that either both statements are

true or both are false. In this case, we say that statements A and B are equivalent.

The words and, or, and not are used with a precise mathematical meaning that

does not always coincide with English usage. It is easy to be tripped up by these

changes in meaning; be careful. “Not A” is the negation of the statement A. So

“not A” is true if and only if A is false. To say that “A and B” is true, we mean that

both A is true and B is true. On the other hand, “A or B” is true when at least one

is true, but both being true is also possible. For example, the statement “if A or C,

then B’ means that if either A is true or C is true, then B is true.

Consider these statements about an integer n:

D. n is even.

E. n is a multiple of 4.

F . There is an integer k so that n = 4k + 2.

The statement “not F” is “there is no integer k so that n = 4k + 2.” The statement

“D and not F ” says that “n is even, and there is no integer k so that n = 4k + 2.”

One can easily check that this is equivalent to statement E. Here are some valid

statements:

(1) D if and only if (E or F ).

(2) If (D and not E), then F.

(3) If F , then D.

In the usual logical system of mathematics, a statement is either true or false,

even if one cannot determine which is valid. A statement that is always true is a

tautology. For example, “A or not A” is a tautology. A more complicated tautology

known as modus ponens is “If A is true, and A implies B, then B is true.” It is

more common that a statement may be true or false depending on the situation (e.g.,

statement D may be true or false depending on the value assigned to n).