Need a Summer Textbook for Undergraduate Analysis Review?

[PieceOfPi]

I just finished taking a year-long undergraduate analysis sequence. The texts I used for this sequence were the first eight chapters of Rudin's PMA (everything before multivariable calculus) and Spivak's Calculus on Manifolds, as well as a little bit of my professor's lecture notes on differential forms and Green's, Gauss's, and Stoke's theorems. I am also thinking of taking the graduate-level analysis sequence (Text: W. Rudin's "Real and Complex Analysis"). Whether I take the graduate-level sequence or not, I think it would be a good idea for me to review and do more problems from the undergraduate analysis sequence again, and I was wondering if there is any good textbook for me to read over this summer. I can certainly read Baby Rudin again, but I also heard a good stuff about C. Pugh's "Real Mathematical Analysis," and I was wondering if this is an appropriate text for me to read over this summer. Let me know if there is any other good textbook.

Thanks

 

[Landau]

Pugh certainly is nice, with a lot of problems. Again (as in your other thread), I'd like to suggest Knapp, namely his Basic Real Analysis, although it is more advanced than undergraduate. Other suggestions: Carothers' Real Analysis, Loomis & Sternberg's Advanced Calculus (free from www), Lang's Undergraduate Analysis, Apostol's Mathematical Analysis, Zorich's Analysis I,II.

 

[PieceOfPi]

Thanks for your suggestion again, Landau! From looking at Knapp's table of contents, I might enjoy that almost everything I learned this year is in the first three chapters, and goes straight into more advanced materials in the later chapters. I might also like the fact that book has hints to the problems at the end of the book, which might be great for self-studying. And it looks like Knapp's Basic Real Analysis is in my library, so I will probably get it pretty soon, and I'm pretty excited about that!

More suggestions are welcome.

 

[Landau]

The books by Knapp (Basic+Advanced Algebra, Basic+Advanced Analysis) are really great. They are extremely clear, have hints and solutions to all exercises, and contain about everything you will encounter in undergraduate and graduate school about algebra and analysis. Actually, as he states in the preface, he wrote them with the goal in mind to provide young mathematicians the background which will be assumed in conferences.

Another suggestion is https://www.amazon.com/dp/3540438734/?tag=pfamazon01-20 by Jost.

 

[rezeew]

for your question! It sounds like you have a strong foundation in analysis and are looking to review and deepen your understanding over the summer. I understand the importance of solidifying your understanding of foundational concepts before moving on to more advanced material.

Based on your background, I would recommend continuing with Rudin's "Real and Complex Analysis" as it builds upon the material you have already covered in his previous texts. However, if you are looking for a different perspective or additional practice problems, Pugh's "Real Mathematical Analysis" is a highly regarded text that covers similar material.

In addition, I would also recommend checking out "Principles of Mathematical Analysis" by Walter Rudin (the author of PMA) as it is a classic text in undergraduate analysis and may provide a different approach to the material you have already covered.

Ultimately, the best textbook for you will depend on your personal learning style and goals. I would suggest browsing through the different texts and seeing which one resonates with you the most. Good luck with your summer studies!