I hear that libgen.io is a useful resource for books...
Ask HN: Resources to learn real analysis?
21–30 of 105 posts
Re: Ask HN: Resources to learn real analysis?
#22Good question ... I haven't seen a great course on real analysis on Youtube/MOOC platforms. If anyone has recs, I would also be interested.
For sure, there’s only so much a lecture in RA can achieve. Ultimately the student has to write rigorous proofs, and lots of them.
Re: Ask HN: Resources to learn real analysis?
#23I worked through 9 chapters of Apostol.
Re: Ask HN: Resources to learn real analysis?
#24Walter Rudin's book on Real Analysis is normally considered as classic in this area. However, you can also take a look at Mathematical Analysis by Tom M. Apostol, which helps in developing good intuition about the subject.
Re: Ask HN: Resources to learn real analysis?
#25Rudin's classic texts are a great resource.
This is how I learned it! (Baby Rudin, anyway.) I wonder if this is still the go-to for undergraduate classes --- does anyone know?
Yes, last year at my university we used Baby Rudin for the undergraduate real analysis sequence, and Tao's books for the graduate level sequence.
Re: Ask HN: Resources to learn real analysis?
#26As others have mentioned, Tao has fantastic resources, and his blog might be worth looking at. Rudin is a classic. I studied real analysis through “Advanced Calculus” by Fitzpatrick which has some great material.
For more advanced analysis (esp. functional analysis) I would look at Kreyszig or Hunter & Nachtergaele.
The best way to prepare imo is to just do proofs between now and the start of the course. Try to find practice proof problems online and see if you can do them or find an entry-level book on discrete math. Problem-Solving Strategies by Engel is a good but slightly more advanced book for a beginner.
Re: Ask HN: Resources to learn real analysis?
#27I don’t have any resources for analysis directly to recommend that haven’t already been said, but there’s some good videos of Calculus by 3blue1brown on YouTube called The Essence of Calculus [1]. They are really well made and explain Calculus in a way that you get an intuitive feel for it. It may be helpful to learning analysis to understand Calculus really well, but I’ve never taken analysis so I can’t say for sure…
Yes, 3Blue1Brown is amazing to get the idea behind and have revelation. At least in my case.
Re: Ask HN: Resources to learn real analysis?
#28I'm probably in a very slim minority of people who took their first topology course before they took their first real analysis course, and I think my understanding of both topics improved as a result of taking them in this order.
there's no way to actually avoid epsilon delta arguments in real analysis, but it's helpful to know that there is a more "intuitive" way of thinking about continuity (although admittedly it's a little weird when you first encounter it), that requires a lot less algebraic magic.
Anyway, there's more to real analysis than the topology of the real numbers, but I think it's a great starting place.
Re: Ask HN: Resources to learn real analysis?
#29Without criticizing any other people’s recommendations on this thread, I think it’s easy for people who have been doing proof-based math for a few years to recommend books that look clear and easy to them now, without remembering the time, effort and other support (e.g. great teacher and classmates) that may have been necessary to make use of that book a good experience.
Or maybe they are just way smart than me? :)
Either way, when considering possible books to use, I would ask the following question - is there a chance this is “too easy” to use / read, while still claiming to be about analysis? (I.e. calculus books fail this test because they don’t say they are about analysis). Then start with the easiest one unless there are really good reasons not to.
My own specific advice would be:
1) make sure you have had practice with proof based math before. If not, or you need the practice, get a copy of chartrand’s “introduction to mathematical proof” and do some exercises from the first 10 chapters. If you can do them easily, move onto analysis, if not, work through those 10 chapters first.
2) The book I personally like best for self-study is Abbott, “Understanding analysis” particularly if you can get the solutions manual, I think the explanations of the proofs are very good.
3) I would also recommend Lara Alcock’s book “How to think about analysis”, which is NOT a textbook, but has a lot of useful information and advice on how to learn analysis.
Also, obvious but worth repeating, if you are taking less than one hour per page to get through an analysis text, or don’t have pencil and paper in hand while going through the book, “ur doing it wrong” :)
Re: Ask HN: Resources to learn real analysis?
#30Coursera's 'Introduction to Mathematical Thinking' is a great starter course for real Analysis https://www.coursera.org/learn/mathematical-thinking