Basic Analysis: Introduction to Real Analysis
Jirí Lebl, Oklahoma State University
Pub Date: 2016
Conditions of Use
This text covers all the standard material for a senior level undergraduate (or master level) real analysis class. It start with basics set theory read more
This text covers all the standard material for a senior level undergraduate (or master level) real analysis class. It start with basics set theory and the real numbers. Then it develops supremum and infimum of bounded sets; limit, limsup, liminf for sequences; series of numbers and functions; continuity and uniform continuity of functions, derivative of functions, Riemann integral and improper integrals, basics of metric spaces including connectedness, compactness set and continuity of functions between metric spaces. Finally it ends with a proof of fixed point theorem. The text covers all the main theorems (such as mean value theorem, intermediate value theorem, Heine-Borel theorem, Bolzano-Weierstrass theorem, Dini’s theorem) one would expect to be covered in this area. It includes a reasonable number of problems and examples. The text provides an effective index at the end.
I did not see anything inaccurate.
I don't expect the text to be obsolete any time soon as this is an advanced math textbook.
The material is presented in a clear fashion and should be very readable by students.
The style of the book is very consistent throughout the book.
It is as modular as one can expect of a book on this subject. Since everything numbered consistently one can easily find the results needed from the previous chapters.
The material is presented in a logical order. Definitions, theorems, examples, exercises, etc are all numbered in a consistent manner. The book is written in a little bit informal language but that is not a shortcoming.
In general, the interface of this book is very typical of an advanced math textbook. All the statements are numbered and hyperlinked making navigation very easy.
Other then few typos I did not see any grammatical errors.
This is not an issue for graduate level mathematics books and this book is no exception.
This book has been approved by the American Institute of Mathematics Open Textbook Initiative. I have used it twice in my classes and have been very happy about it. The author maintains an errata on his website and has been updating the text regularly. I suggest using the latest edition that can be obtained from the authors website.
The textbook covers everything I would want to cover in a course at this level and some more. A teacher can use this book as the sole text of an read more
The textbook covers everything I would want to cover in a course at this level and some more. A teacher can use this book as the sole text of an introductory analysis class and skip the last two chapters for a slower class.
I found a few minor typographical errors but no serious problems.
This is exactly the material that should be covered in an introductory analysis class and it will remain relevant up to date for a long time.
The material is presented in a clear, concise manner. Every term is adequately defined.
The author does a great job of remaining consistent with terminology throughout the book.
It is difficult to write a highly modular textbook with this material since many results depend on earlier ones. However, this book does as good of a job as could be expected by breaking down the material into reasonable sections and maintaining a consistent numbering scheme throughout the book. When earlier results need to be referenced, they are easy to find.
I am glad that the book opens with set theory. The lack of such material can be a shortcoming of some analysis texts. The chapters are presented in a logical order, with all of the material building on previous results.
I had no trouble reading this book or finding results. I appreciate the use of a consistent numbering scheme for results throughout the textbook.
I found a few typographical errors, but otherwise everything was fine.
This material should be appropriate in any classroom where this subject is being taught. I saw nothing that seemed culturally insensitive.
This is a great textbook for introductory analysis, and I expect that I will use it the next time I teach the subject.
Table of Contents
- Chapter 1. Real Numbers
- Chapter 2. Sequences and Series
- Chapter 3. Continuous Functions
- Chapter 4. The Derivative
- Chapter 5. The Riemann Integral
- Cjhapter 6. Sequences of Functions
- Chapter 7: Metric Spaces (New in November 18th 2011 edition)
About the Book
This free online textbook (e-book in webspeak) is a one semester course in basic analysis. This book started its life as my lecture notes for Math 444 at the University of Illinois at Urbana-Champaign (UIUC) in the fall semester of 2009, and was later enhanced to teach Math 521 at University of Wisconsin-Madison (UW-Madison). A prerequisite for the course is a basic proof course. It should be possible to use the book for both a basic course for students who do not necessarily wish to go to graduate school, but also as a first semester of a more advanced course that also covers topics such as metric spaces.
About the Contributors
Jirí Lebl, Assistant Professor, Department of Mathematics, Oklahoma State University.