Analysis on Selecting the Real Analysis textbooks

[bacte2013]

Dear Physics Forum friends,

I am a sophomore in US with double majors in mathematics and microbiology. I am interested in self-studying the real analysis starting now since it will help me on my current research on computational microbiology, prepare for upcoming math research (starting on this Fall) on the analytic number theory, and prepare for the real analysis course I will take on Fall and Putnam competition. I just finished "Calculus with Analytic Geometry" by G. Simmons, "How to Prove It" by Daniel Velleman, and "How to Solve It" by G. Polya. I also read some portions of Apostol's Calculu Vol.I to get a deeper view on the calculus theories. I was originally planned to read Apostol's Calculus Vol.I and Spivak's Calculus first, but I think it would be a better idea to start with real analysis since it covers all ideas in those "advanced calculus" textbooks and much more. My current plan is to start with one "dumbed-down" real analysis textbook and one "comprehensive, detailed, and intermediate" textbook, and advance into Rudin's PMA (required textbook for my real analysis course) starting on Summer and use it in accordance with other real analysis textbooks. Could you help me on selecting one book from each category?

Elementary Real Analysis textbooks:
**Elementary Analysis: The Theory of Calculus (Kenneth Ross)
**Understanding Analysis (Steven Abbott)
**The Way of Analysis (Robert Strichartz)
**Real Mathematical Analysis (Charles Pugh)

Intermediate, detailed Real Analysis textbooks:
**Mathematical Analysis (Tom Apostol)
**Undergraduate Analysis (Serge Lang)
**Introduction to Real Analysis (Bartle, Sherbert)
**Elements of Real Analysis (Bartle, Sherbert)
**Mathematical Analysis I (Vladimir Zorich)Thank you very much for your time, and I look forward to your advice!

PK

 

[jedishrfu]

What about Rudin:

https://www.amazon.com/dp/0070542341/?tag=pfamazon01-20

and Baby Rudin:

https://www.amazon.com/dp/0070856133/?tag=pfamazon01-20

 

[bacte2013]

I am trying to choose either "Undergraduate Analysis" by Serge Lang or "Mathematical Analysis" by Tom Apostol along with "The Way of Analysis" by Robert Strichartz, and later use Rudin's PMA once I am in midway of studying those books. Should I buy both Lang and Apostol?

 

[jbunniii]

Don't underestimate Spivak's "Calculus." Despite its name, it certainly qualifies as an introduction to real analysis. I think you would find it more challenging (in a good way) than the book by Ross, for example. The exposition is rigorous but lively and fun to read, and the exercises are great. Spivak would be my strong recommendation unless you are already very comfortable with rigorous calculus (epsilon-delta proofs) on the real line.

If you already know Spivak-level calculus, then you could proceed straight to Rudin if desired. But if you want an alternative, then among the ones listed, I would go with Pugh's. You have seriously miscategorized it by filing it in the introductory section with Ross; it's probably the hardest book in your list. It is at the same level as Rudin. It has a more talkative and informal exposition (not my preference, but many people like it), and it has an amazing collection of challenging exercises, including Berkeley prelim exam problems.