Self teaching analysis and topology

[diegzumillo]

TL;DR Summary

Give me suggestions on books, online courses, some words of wisdom etc.

Hi,
I am a physicist interested in going through a basic analysis course. The real line, open sets, that whole thing. On my own, so I need a good selection of bibliography, ranging from those that are good references but too dense to actually read to those that are very pedagogical but tend to be lacking in rigor. Any advice or other suggestions is also welcome.

 

[S.G. Janssens]

There are probably quite some old topics about this in the textbook section, so you should look there, too. If your school offers access to AMS reviews, I would browse those, too. Good introductions are, for example, Bartle and Sherbert for analysis and Croom for metric and point-set topology.

On my own, so I need a good selection of bibliography, ranging from those that are good references but too dense to actually read to those that are very pedagogical but tend to be lacking in rigor.

References are usually not introductions (there are exceptions, such as the Hitchhiker's book by Aliprantis and Border). If you would like to learn these subjects, I would stay away from books that lack rigor. In my opinion, an introduction to analysis and/or topology should be (and can be) both pedagogical and rigorous.

 

[FactChecker]

I have always been a big fan of the Schaum's Outline series of books. They all have many worked examples and exercises. There are many of them and you can pick the ones that suit your needs.

 

[S.G. Janssens]

I have always been a big fan of the Schaum's Outline series of books. They all have many worked examples and exercises. There are many of them and you can pick the ones that suit your needs.

Out of curiosity, are these also good as self-contained introductions to a certain subject? It has always been my impression that they are mostly supplemental, or even mainly "cramming-aids" for tests, but probably that is unjustified.

 

[FactChecker]

Out of curiosity, are these also good as self-contained introductions to a certain subject? It has always been my impression that they are mostly supplemental, or even mainly "cramming-aids" for tests, but probably that is unjustified.

Thanks. That might be a fair statement. I only used them as supplemental practice material. I don't know how well they would work as introductory material. They do have explanatory text, but I don't know how good it is. I will look through some and see what I think.

 

[martinbn]

Shilov's books
"Elementary Real and Complex Analysis"
"Mathematical Analysis: A Spacial Course"
"Elementary Functional Analysis"

 

[diegzumillo]

If you would like to learn these subjects, I would stay away from books that lack rigor. In my opinion, an introduction to analysis and/or topology should be (and can be) both pedagogical and rigorous.

I hear you. At least in physics, whenever self teaching any subject (and even when taking a course) I rarely manage with a single source. And when choosing alternative sources I try to pick different approaches to the same subject, that is why I asked for those extremes. But I am in unfamiliar territories here.

As for everyone else, thanks for the recommendations! I am taking note of them all.

And my own contribution: I found a really nice set of lectures on youtube by Professor Francis Su. Comes with lecture notes and everything. Link to video 2.

 

[mathwonk]

I started out in topology taking a lecture course from Raoul Bott, who was, as you may know, one of the greatest topologists of the 20th century. He defined a continuous map as one such that the inverse image of an open set is open, and a homeomorphism as a continuous map with a continuous inverse. Then he asked if we thought the sphere was homeomorphic to the torus, and i said yes. He asked me to start him off on a homeomorphism, but I could not.

One can learn several things from this: 1) that definition of homeomorphism is not very useful on its own; 2) one should always examine examples; 3) to show a space is homeomorphic to another, one should first at least try to find a map from one to the other; 4) a book that does not make it clear rather early on that a sphere is not homeomorphic to a torus is not too helpful; 5) I was totally out of my depth in that course, and I at least got that message.

FWIW: The key idea in topology for distinguishing non - homeomorphic spaces, is connectivity. A sphere is disconnected by removing any closed curve, but a torus is not, hence they are not homeomorphic.

None of this is Bott's fault, who was also one of the best teachers of topology I ever encountered. But a kindergartener should not be in a college class.

by the way, the book that finally got me started in a good way, i.e. by providing some intuition, was by Alexandroff. One of the best English language authors of topology books is William S. Massey. Anything by him is good.

 

[Infrared]

A sphere is disconnected by removing any closed curve

I'll just point out that this is far from obvious, as it is essentially the content of the Jordan curve theorem!

 

[MidgetDwarf]

Analysis was one of those math courses that was difficult to learn from books, compared to say Modern Algebra books. To my eyes, it seems that Analysis is more difficult to learn. Since I want to be an Analyst or a Geometer, I need to practice more Analysis. End of my rant.

The book that I learned some analysis from and where ideas started to click was Abbot: Understanding Analysis. Clear explanations, although it only focuses in R, and no mention of metric spaces. At least the parts I read. So one must read a more "rigorous" book. The good news is that Pugh: Real Mathematical Analysis, covers these shortcomings. I am only on page 20, but I like the book so far. It compliments Abbot nicely..

Although I found the beginning pages of construction of R using Dedekind cuts a bit hard to follow for me, so I decided to skip. I think Bloch: Real Number and Real Analysis, covers this nicely. I wish that Pugh would have actually shown the reader why {x in R : x<1}|{x in R : x>=1} was a Dedekind Cut, and not just mention so. Bloch actually walks you through the construction of the Reals with a few different approaches.

What I liked from Pugh, was his proof that if a sequence is Cauchy then it also is a convergent sequence. Which is different from the proof Abbot provides. Both authors prove first that a Cauchy sequence is bounded. But Abbot uses the idea of subsequences, motivated before Cauchy sequence, and uses the theorem that every bounded sequence has a convergent subsequence (Bolzano- Weierstrass Theorem). Instead, Pugh does it via construction of a set with a certain condition. Two nice and distinct proofs.

I would say both books compliment each other well, but I would not recommend only buying Pugh. There is a lot of stuff so far, that is assumed to be common knowledge for the reader, and not many examples. Ie., what a sequence is and examples of a sequence. He does say it, but its not entirely obvious. But, I think this was done because the audience for the book was for Berkly students.

Buy both.