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!
Best Answer
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.