Problems and Solutions for Undergraduate Real Analysis II

210 Pages·2019·3.956 MB·English

by  Kit-Wing Yu

Problems and Solutions for
Undergraduate Real Analysis II
by Kit-Wing Yu, PhD
[email protected]
Copyright c 2019 by Kit-Wing Yu. All rights reserved. No part of this publication may be
(cid:13)
reproduced,storedinaretrievalsystem,ortransmitted,inanyformorbyanymeans,electronic,
mechanical, photocopying, recording, or otherwise, without the prior written permission of the
author.
ISBN: 978-988-78797-6-3 (eBook)
ISBN: 978-988-78797-7-0 (Paperback)
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About the author
Dr. Kit-Wing Yu received his B.Sc. (1st Hons), M.Phil. and Ph.D. degrees in Math. at the
HKUST,PGDE (Mathematics) at the CUHK.After his graduation, hehas joined United Chris-
tian College to serve as a mathematics teacher for at least nineteen years. He has also taken
the responsibility of the mathematics panel since 2002. Furthermore, he was appointed as a
part-time tutor (2002 – 2005) and then a part-time course coordinator (2006 – 2010) of the
Department of Mathematics at the OUHK.
Apart from teaching, Dr. Yu has been appointed to be a marker of the HKAL Pure Mathe-
matics and HKDSE Mathematics (Core Part) for over thirteen years. Between 2012 and 2014,
Dr. Yu was invited to be a Judge Member by the World Olympic Mathematics Competition
(China). In the area of academic publication, he is the author of four books
A Complete Solution Guide to Real and Complex Analysis I.
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Problems and Solutions for Undergraduate Real Analysis I.
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Mock Tests for the ACT Mathematics.
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A Complete Solution Guide to Principles of Mathematical Analysis.
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Besides, he has published over twelve research papers in international mathematical journals,
includingsome well-known journalssuchas J.Reine Angew. Math., Proc. Roy. Soc. Edinburgh
Sect. A and Kodai Math. J.. His research interests are inequalities, special functions and
Nevanlinna’s value distribution theory.
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Preface
This book “Problems and Solutions for Undergraduate Real Analysis II” is the continuum of
the first book “Problems and Solutions for Undergraduate Real Analysis I”. Its aim is the same
as its first book: We want to assist undergraduate students or first-year students who study
mathematics in learning their first rigorous real analysis course.
“The only way to learn mathematics is to do mathematics.” – Paul Halmos. My learning
and teaching experience has convinced me that this assertion is definitely true. In fact, I believe
that “doing mathematics” means a lot to everyone who studies or teaches mathematics. It is
not only a way of writing a solution to a mathematical problem, but also a mean of reflect-
ing mathematics deeply, exercising mathematical techniques expertly, exchanging mathematical
thoughts with others effectively and searching new mathematical ideas unexpectedly. Thus I
hopeeveryonewhoisreadingthisbookcanexperienceandacquiretheabovebenefitseventually.
The wide variety of problems, which are of varying difficulty, include the following topics:
Sequences andSeriesof Functions, ImproperIntegrals, LebesgueMeasure, LebesgueMeasurable
Functions, Lebesgue Integration, Differential Calculus of Functions of Several Variables and
Integral Calculus of Functions of Several Variables. Furthermore, the main features of this book
are listed as follows:
The book contains 226 problems, which cover the topics mentioned above, with detailed
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and complete solutions. Particularly, we include over 100 problems for the Lebesgue inte-
gration theory which, I believe, is totally new to all undergraduate students.
Each chapter starts with a brief and concise note of introducing the notations, terminolo-
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gies,basicmathematicalconceptsorimportant/famous/frequentlyusedtheorems(without
proofs) relevant to the topic.
Three levels of difficulty have been assigned to problems:
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Symbol Level of difficulty Meaning
⋆ Introductory These problems are basic and every
student must be familiar with them.
⋆ ⋆ Intermediate The depth and the complexity of the
problems increase. Students who target
for higher grades must study them.
⋆ ⋆ ⋆ Advanced These problems are very difficult and
they may need some specific skills.
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Colors are used frequently in order to highlight or explain problems, examples, remarks,
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main points/formulas involved, or show the steps of manipulation in some complicated
proofs. (ebook only)
If you find any typos or mistakes, please feel free to send your valuable comments or opinions
to
[email protected]
Any updated errata of this book or news about my new book will be posted on my new website:
https://sites.google.com/view/yukitwing/
Kit Wing Yu
July 2019
List of Figures
10.1 An example of pointwise convergence. . . . . . . . . . . . . . . . . . . . . . . . . 2
10.2 An example of uniform convergence. . . . . . . . . . . . . . . . . . . . . . . . . . 2
15.1 The Inverse Function Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
16.1 The subinterval in R2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
16.2 The outer and the inner Jordan measures. . . . . . . . . . . . . . . . . . . . . . . 171
16.3 The mapping φ: E D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
→
16.4 The mapping φ: E R. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
→
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List of Figures viii
Contents
Preface v
List of Figures vii
10 Sequences and Series of Functions 1
10.1 Fundamental Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
10.2 Uniform Convergence for Sequences of Functions . . . . . . . . . . . . . . . . . . 6
10.3 Uniform Convergence for Series of Functions . . . . . . . . . . . . . . . . . . . . . 15
10.4 Equicontinuous Families of Functions . . . . . . . . . . . . . . . . . . . . . . . . . 23
10.5 Approximation by Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
11 Improper Integrals 33
11.1 Fundamental Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
11.2 Evaluations of Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
11.3 Convergence of Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 42
11.4 Miscellaneous Problems on Improper Integrals . . . . . . . . . . . . . . . . . . . . 50
12 Lebesgue Measure 57
12.1 Fundamental Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
12.2 Lebesgue Outer Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
12.3 Lebesgue Measurable Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
12.4 Necessary and Sufficient Conditions for Measurable Sets . . . . . . . . . . . . . . 78
13 Lebesgue Measurable Functions 83
13.1 Fundamental Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
13.2 Lebesgue Measurable Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
13.3 Applications of Littlewood’s Three Principles . . . . . . . . . . . . . . . . . . . . 96
14 Lebesgue Integration 105
14.1 Fundamental Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
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