Advanced Real Analysis Book: Recommendations for Next Level Learning

[SticksandStones]

Hello all. Last semester I took a real analysis class using this book:
https://www.amazon.com/dp/038790459X/?tag=pfamazon01-20

Well, we finished this book and now I'd like to continue on and learn what comes next (if you will).

Any recommendations for a more advanced analysis book?

EDIT: I was thinking this: https://www.amazon.com/dp/0387948414/?tag=pfamazon01-20

 

[n!kofeyn]

I guess it depends on what your interested in. There are many paths in analysis to take once you complete a basic real analysis course. I've tried to list many of these paths and the relevant books that are of course my favorites. :)

Mathematical Analysis by Tom Apostol
This is if you want to continue along the standard real analysis path. It is very comprehensive, but a little dry.

Introductory Functional Analysis with Applications by Erwin Kreyszig
If you want to learn functional analysis, then this book is the best introduction there is and has a minimal amount of prerequisites, which you've already completed.

Advanced Calculus: A Differential Forms Approach by Harold Edwards
This is a great and unique introduction to the analysis of differential forms.

Lebesgue Integration and Measure by Alan Weir
This is probably the most basic and intuitive introduction to Lebesgue integration theory. It isn't as rigorous as the above texts or other texts on Lebesgue theory, but it is a good book for someone at your level who has just completed a good course in analysis.

Visual Complex Analysis by Tristan Needham
Complex Variables by George Polya (an old book but available used on Amazon)
There are many good texts on complex analysis, and the subject is conquerable by anyone equipped with a good course in real analysis. I really liked reading through the book by Polya. Needham's book is very unique, but I haven't read it. Like I said though, there are many complex analysis books out there.

 

[SticksandStones]

Thank you for that indepth response, twas quite helpful :)

I decided to order Advanced Calculus: A Differential Forms Approach, mostly because it was cheap and had a preview on Amazon. The worst that can happen is I learn something new!

Thanks again!

 

[n!kofeyn]

Okay, no problem! I'm actually a little surprised you ordered that one, as it usually gets overlooked. It's not like other texts, but hopefully you enjoy it. He includes some very cool discussions and doesn't follow the same rubric as the usual textbooks. Also, the nice thing about it is that it's a subject you're not likely to encounter during your undergraduate studies except possibly briefly at the end of a vector calculus course. I didn't learn about differential forms until my first year in graduate school.

I think it would be neat if a couple months down the line, you came back here and posted your thoughts about it.

 

[SticksandStones]

I'll remember to do that. :)

 

[Landau]

Actually, I think Lang's Undergraduate Analysis is one of the best choices. There are lots of other good books on analysis, of course, and as n!kofeyn said there are many ways to proceed. To name a few others that I like: Carothers - Real Analysis, Knapp - Basic Real Analysis, Loomis & Sternberg - Advanced Calculus.

 

[dslowik]

I just love Pugh's Real Mathematical Analysis..