# Real Analysis Texts: Bartle Analysis Reviews

• Cod
In summary, Robert Bartle's "Introduction to Real Analysis" is a more rigorous text than Rudin's "Baby Analysis." It is aimed about midway between Spivak's "Calculus" and Rudin's "Baby Analysis," and has more sophisticated material in terms of topology, Fourier series, and integration. It is also a great book for problem solving, with many challenging projects.

#### Cod

I'm taking Real Analysis I this semester and looking to purchase a more rigorous text in the next few months. I know everyone swears by Rudin, but I was curious about everyone's thoughts on Robert Bartle's Introduction to Real Analysis (new edition just released in January 2011) and/or The Elements of Real Analysis.

The Elements of Real Analysis is the text that was used in my first analysis course, 20 years ago. It was a semester-long honors course that focused on the real line (one dimension). We covered a large portion of the book, and I spent many hours with it; my copy is in pieces at this point.

It's an outstanding book, in my opinion. In terms of mathematical maturity, it is aimed about midway between Spivak's Calculus and Rudin's baby analysis book. It isn't as terse as Rudin's or as conversational as Spivak's.

A good illustration of the relative level of sophistication is the treatment of topology in these three books.

Spivak develops just enough topology to prove a few theorems about continuous functions on closed intervals, without (as I recall) ever defining such things as compactness, connectedness, or even general (non-interval) open or closed sets.

Bartle does cover the standard elementary topology used in analysis: compactness, connectedness, Heine-Borel, equicontinuity, etc., all in the special case of R^n. He doesn't talk about general metric spaces at all, except in an exercise or two.

Rudin works with metric spaces and specializes the results to R^n.

Both Bartle and Rudin present the Riemann-Stieltjes integral, with the Riemann integral as a special case. Rudin has a rather perfunctory chapter at the end of the book covering the Lebesgue theory; Bartle doesn't.

Bartle has a generous amount of material on Fourier series, perhaps about as much as you can do with Riemann integrals.

Bartle has a lot of really good problem sets, both more generous in number and less difficult than Rudin's. Most of these are extensions or applications of the theory covered in the text. They are, I think, the "right" level of challenge for undergraduate analysis - substantial enough that you will come away with a good understanding of the material, but not so hard as to be unreasonable. I spent many hours each week on the problem sets, but in the end was usually able to do all of them.

In many of the chapters, Bartle also has a set of "projects", which are a series of exercises developing a new topic. These are really great, a highlight of the book. Our professor assigned several of them throughout the semester.

For example, in chapter 30 ("Existence of the Integral"), he has four projects in addition to the regular problem set. The first project (in 7 parts) develops the logarithm function and its properties starting from the integral definition. The second project (in 6 parts) does the same thing with sine and cosine. The third project develops the Wallis product in 5 parts. The fourth does the same with the Stirling formula, in 6 parts.

Overall, I think very highly of Bartle's Elements book. If you forced me to get rid of all of my undergraduate analysis books except one, I would keep baby Rudin because it is less cluttered with pedagogy and serves as a better reference - and for the same reason, I don't especially recommend Rudin as a good first book to learn from. If you let me keep one more, it would be Bartle.

I have not read Bartle's "Introduction to Real Analysis" book, but my understanding is that it covers less material, and at a slower pace, than "Elements." However, in more recent years, Bartle has become a big advocate of the Henstock-Kurzweil, aka generalized Riemann, integral, so I'm not surprised to see, looking at the Amazon preview, that he covers that integral in the new edition of "Introduction." (It is not covered in the older "Elements.")

He has also written a full-length book on the Henstock-Kurzweil integral, "A Modern Theory of Integration."

Finally, I will also mention that his wonderful, albeit very expensive per page, slender volume "The Elements of Integration and Lebesgue Measure" is well worth reading when/if you decide to study Lebesgue integration.

Last edited:
Actually, looking through the copy of Elements of Real Analysis in front of me, it doesn't look like it includes anything on Fourier Series. There's nothing mentioned in the table of contents, and Fourier isn't in the index.

It's a shame, because I've actually been looking for a book that can help supplement Rudin's coverage of Fourier Series.

Thanks for the great breakdowns of the texts. I'm planning on getting one of his books in the next week.

tcbh said:
Actually, looking through the copy of Elements of Real Analysis in front of me, it doesn't look like it includes anything on Fourier Series. There's nothing mentioned in the table of contents, and Fourier isn't in the index.

It's a shame, because I've actually been looking for a book that can help supplement Rudin's coverage of Fourier Series.

Maybe we have different editions. I have the 2nd edition, copyright 1976 (ISBN 047105464X). Chapter 37 is "Series of Functions," Chapter 38 is "Fourier Series." Fourier is in the index.

qspeechc said:
I have not read the books, but you might try the Princeton Lecture Series by Elias Stein (have a look at them if the library has a copy):
https://www.amazon.com/dp/069111384X/?tag=pfamazon01-20
https://www.amazon.com/dp/0691113858/?tag=pfamazon01-20
https://www.amazon.com/dp/0691113866/?tag=pfamazon01-20
I think the books depend on material in the previous books, however.

I've read part of the Fourier Analysis book - it's very good.

By the way, it seems that the long-promised Book 4 is finally coming out next month:

https://www.amazon.com/dp/0691113874/?tag=pfamazon01-20

Last edited by a moderator:
jbunniii said:
Maybe we have different editions. I have the 2nd edition, copyright 1976 (ISBN 047105464X). Chapter 37 is "Series of Functions," Chapter 38 is "Fourier Series." Fourier is in the index.
Yes, the 1964 edition of "Elements" stops at Ch 28 - Series of Functions. Fourier is not mentioned in the index. It is nice that the newer edition is actually improved!

Maybe we have different editions. I have the 2nd edition, copyright 1976 (ISBN 047105464X). Chapter 37 is "Series of Functions," Chapter 38 is "Fourier Series." Fourier is in the index.

Ah, yes. I do have the 1964 edition. I should go get the 1976 edition then!

## 1. What is Real Analysis?

Real Analysis is a branch of mathematics that deals with the study of real numbers, their properties, and the functions defined on them. It is considered to be a more advanced version of calculus, focusing on rigorous mathematical proof and the underlying theory behind calculus concepts.

## 2. What are some popular Real Analysis textbooks?

Some popular Real Analysis textbooks include "Introduction to Real Analysis" by Bartle and Sherbert, "Principles of Mathematical Analysis" by Rudin, and "Real Analysis" by Royden and Fitzpatrick.

## 3. What can I expect to learn from a Real Analysis course?

In a Real Analysis course, you can expect to learn about limits, continuity, differentiation, integration, and series of real-valued functions. You will also learn about the theoretical foundations of calculus, such as the completeness of the real numbers and the concept of convergence.

## 4. How difficult is Real Analysis compared to other math courses?

Real Analysis is considered to be a challenging course, as it requires a strong understanding of calculus and mathematical proof techniques. It is typically taken by upper-level undergraduate or graduate students in mathematics or related fields.

## 5. Are there any recommended resources for studying Real Analysis?

In addition to textbooks, there are many online resources available for studying Real Analysis, such as lecture notes, practice problems, and video lectures. Some popular websites for Real Analysis resources include Khan Academy, MIT OpenCourseWare, and Paul's Online Math Notes.