Real Analysis advanced book suggestion

Question

I want to self study real analysis. So far, finished the first seven chapters of Baby Rudin (up to and including sequences and series of functions) and now want to proceed into more advanced books.

I have couple options, including

• Stein&Shakarchi
• Folland
• Royden
• Rudin's Real&Complex Analysis
• Kolmogorov-Fomin

Among these five (also happy to hear if you have further recommendations) which are more accessible and has better treatment of the material? I'm especially thinking among first three, so if there would be a comparative answer for the first three books, I would be really happy. Any help is appreciated. Thank you!

Answer 1

The first two books you listed are excellent, and it may be worth reading the two together. Stein's book (at least in the first three chapters) focuses on Lebesgue measure in $\mathbb{R}^n$, while Folland takes a more general, abstract approach. It can be useful to have a concrete special case to think about when learning the general theory. There's a similar contrast in the functional analysis sections of the two books: Stein focuses on the simpler case of Hilbert spaces but goes into more depth, while Folland says more about the more general theory of Banach spaces.

The real analysis portion of big Rudin covers more or less the same ground as Folland. I prefer Folland, but big Rudin is also a good book.

Answer 2

Answer: all of them.

The entire Stein&Shakarchi series is excellent, so if you like their style it's a great addition to your library. Especially if you're later interested in complex analysis, harmonic analysis and analytic number theory. The examples and additional lemmas in their books are a great supplement to your understanding of real analysis.

Royden is an excellent overview of real analysis, providing down-to-earth proofs and examples. It's probably the easiest to read out of the books listed. Make sure to get 3rd edition or higher as earlier ones are riddled with typos.

Rudin's Real&Complex Analysis is a work of art, and I would suggest using it as a supplement to any of the above. His exposition is beautifully terse but concise. The reason I suggest another supplement is its a bit too terse, leaving most pressing questions as an exercise to the reader.

Folland is excellent in his presentation of measure theory, relegated to just 3 chapters, and then moving onto point set topology.

Kolmogorov-Fomin is a more classic text, especially in its notation. It's tiny in frame, but dense on details. Would recommend it as a supplement to any of the above.

One suggestion would be to study real analysis with applications in mind. One example is to simultaneously study advanced probability theory, which will supplement your understanding of sigma algebras and general convergence principles in real analysis. Probability: Theory and Examples by Durrett is exhaustive.

Answer 3

I would go for big Rudin. It will give you a solid base for analysis. Folland is also a god option, the second best one in your list.

There exists other books that presents Analysis with a perspective from Measure theory and Topology:

• Topology approach: Elon Lages Lima - Curse de Analisé Vol I and Vol II.
• Measure Theory approach : (I lost the name, I will update soon)

Also, this one: Delinger - Elements of Real analysis

Answer 4

I’m studying real analysis in Ukraine, so I use a local set of books. The main of them I found at home when I was a child, and still hoping to completely read someday, is a fundamental, very detailed and so huge (it has more than two thousands pages in three volumes) book “Differential and Integral Calculus” by Grigorii Fichtenholz. This is a famous book for our students, and, according to Wikipedia,

The bookwas translated, among others, into German, Chinese, and Persian however a translation to English language has not been done still.

Fichtenholz's books about analysis are widely used in Middle and Eastern European as well as Chinese universities due to its exceptionality of detailed and well-ordered presentation of material about mathematical analysis. Due to unknown reasons, these books do not have the same fame in universities in other areas of the world.

Another book is “Mathematical analysis” by A.Ya. Dorogovtsev, with more modern (1993-4) and brief (only two small volumes with a bit more than six hundred pages in total) approach.