# Basic Analysis: Introduction to Real Analysis

Jirí Lebl, Oklahoma State University

Copyright Year: 2016

Last Update: 2020

Publisher: Jirí Lebl

Language: English

## Formats Available

## Conditions of Use

Attribution-NonCommercial-ShareAlike

CC BY-NC-SA

## Reviews

This book gives a very thorough coverage from set-theoretic prerequisites to difficult questions of the more advanced topics that students need for Real Analysis. The proofs are helpfully detailed. The little tricky parts are not skipped or left... read more

This book gives a very thorough coverage from set-theoretic prerequisites to difficult questions of the more advanced topics that students need for Real Analysis. The proofs are helpfully detailed. The little tricky parts are not skipped or left to the reader. A professor trying to demonstrate rigor to their students will appreciate the choices made by the author. The book has a good index and a comprehensive glossary. The little touches are greatly appreciated.

The book is well-written and has clearly been scrubbed of simple errors. This is a book that I used with much success as a very young instructor who needed a solid base of material. I developed confident that this book had correctly worked examples and proofs.

This is exactly the material that a student needs to see in their first two semesters of introductory analysis. It can be used as a first proof-based course coming after a linear algebra or possibly concurrently. Different people will find it fitting their curriculum differently. But this book provides a very nice two semester course that starts by introducing set theory and induction for the first time and ends with students ready for topology, measure theory, or more advanced calculus.

The clarity of the writing is appreciated. As a proof-based math text of course students will need to be guided through reading it. But the pointers and language are arranged to maximize usability by a faculty person. Because the book has some volume, an instructor can challenge their students to read some sections independently with confidence that the material is arranged clearly enough, obscured only by the usual challenges related to understanding analysis.

This book sticks to its organizing principles. The definition-example-proof-theorem-exercise setup is tried and true.

Sections are broken into subsections in appropriate places. It is appropriately self-referential in a way that is necessary for a book building up a mathematical theory. The way in which some sections are optional is explained on the first page of the introduction.

The text builds up the material in a sensible fashion. Some of the ordering choices are the subject of lively pre-existing lively debates. But all of the organizational choices here are logical and lead to a workable year if taken on order.

I did not notice any interface issues.

I do not remember being troubled by any grammatical errors.

This book is not culturally insensitive or offensive. Books in this subject rarely are.

I recommend this book very highly. I used it with much success as a first proof-based course for sophomore/ junior level students at Pitt. You should give it a try.

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... 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 introductory analysis class and skip the last two chapters for a slower class. 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

- Introduction
- 1 Real Numbers
- 2 Sequences and Series
- 3 Continuous Functions
- 4 The Derivative
- 5 The Riemann Integral
- 6 Sequences of Functions
- 7 Metric Spaces

## 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

### Author

**Jirí Lebl**, Assistant Professor, Department of Mathematics, Oklahoma State University.