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As far as books go, we used "Real Analysis" by Carothers for analysis in undergrad at Caltech and it's one of only two math textbooks (the other being Dummit an

As far as books go, we used "Real Analysis" by Carothers for analysis in undergrad at Caltech and it's one of only two math textbooks (the other being Dummit and Foote for algebra) that I go out of my way to recommend, particularly for self study. It's probably not a sufficient book if you're pursuing a Ph.D. or anything, but I would definitely not start out with Rudin (too little hand-holding, you'll die), and Carothers was amazing.

rsj_hn 5y ago
I recommend anything by George F. Simmons, as well as anything by Russians. Seriously, if it's a Russian sounding author's name, then odds are good that it's well written. They had amazing pedagogy and the EMS series (a joint publication by Springer and the old Soviet publisher VINITI) is first rate. https://www.amazon.com/Encyclopaedia-of-Mathematical-Sciences/dp/B08BC5GZWZ https://www.amazon.com/Encyclopaedia-of-Mathematical-Science...
kxyvr 5y ago
Thanks for providing another suggestion on the real analysis front. Really, I agree with you on the Rudin front. Honestly, I think that real analysis is the largest barrier for people trying to self study applied math. I'll mention here that I actually had a total of five semesters in real analysis all said in done. The first was an intro to real analysis course that used "Introduction to Real Analysis" by Bartle and Sherbert. The second was a more advanced real analysis course that was two semesters long using "Real Analysis and Foundations" by Krantz. When I entered my Ph.D, I was given an initial evaluation and the committee found my real analysis still too weak, which necessitated another two semesters using baby Rudin. Only at that point, did it really take and it's not been an issue since. Anyway, mostly I wanted to echo that I agree with you. I think Rudin is the metric to say good enough, but I was not able to get through it on first go.

rsj_hn 5y ago

I recommend anything by George F. Simmons, as well as anything by Russians. Seriously, if it's a Russian sounding author's name, then odds are good that it's well written. They had amazing pedagogy and the EMS series (a joint publication by Springer and the old Soviet publisher VINITI) is first rate. https://www.amazon.com/Encyclopaedia-of-Mathematical-Sciences/dp/B08BC5GZWZ https://www.amazon.com/Encyclopaedia-of-Mathematical-Science...

kxyvr 5y ago

Thanks for providing another suggestion on the real analysis front. Really, I agree with you on the Rudin front. Honestly, I think that real analysis is the largest barrier for people trying to self study applied math. I'll mention here that I actually had a total of five semesters in real analysis all said in done. The first was an intro to real analysis course that used "Introduction to Real Analysis" by Bartle and Sherbert. The second was a more advanced real analysis course that was two semesters long using "Real Analysis and Foundations" by Krantz. When I entered my Ph.D, I was given an initial evaluation and the committee found my real analysis still too weak, which necessitated another two semesters using baby Rudin. Only at that point, did it really take and it's not been an issue since. Anyway, mostly I wanted to echo that I agree with you. I think Rudin is the metric to say good enough, but I was not able to get through it on first go.