Books for somewhat more advanced real analysis/metric spaces

[AndreasC]

There is a class in my uni called Real Analysis, which covers mostly metric spaces and generally a bunch of slightly more advanced topics than just your basic calculus/multivariable calculus/very basic number theory etc. There are some notes that you can download but for some reason no recommended books so I'm looking for something good.

Here are most of the subjects the class covers to give you a better idea:
Peano axioms, Properties of real and natural numbers, Dedekind cuts, metric space definitions, examples and metrics in vector spaces defined by norms, open and closed subsets of metric spaces, equivalent metrics, countable and uncountable sets, Zorn lemma, metric space completeness, Baire theorem, C[a, b] spaces, Arzela theorem, products of metric spaces, Cantor set etc.

 

[Infrared]

The standard textbook is Rudin's Principles of Mathematical Analysis. I like it, but it's a little terse and can be hard to read on its own. I think it's good when you're also taking a class though. It also has a good range of problems. I don't think it talks about normed vector spaces or Zorn's lemma.

I also like Carothers' book on real analysis. It covers some topics Rudin doesn't and is gentler reading in my opinion.

 

[mathwonk]

You might take a look at Dieudonne's Foundations of Modern Analysis. It's on a rather high level and is demanding but rewarding reading.

 

[AndreasC]

The standard textbook is Rudin's Principles of Mathematical Analysis. I like it, but it's a little terse and can be hard to read on its own. I think it's good when you're also taking a class though. It also has a good range of problems. I don't think it talks about normed vector spaces or Zorn's lemma.

I also like Carothers' book on real analysis. It covers some topics Rudin doesn't and is gentler reading in my opinion.

I checked Rudin and it doesn't look like it covers the material I need. The vast majority of that material has been covered by my Analysis 1, 2 and 3 classes.

Carothers does look like it covers at least some of that.

 

[AndreasC]

You might take a look at Dieudonne's Foundations of Modern Analysis. It's on a rather high level and is demanding but rewarding reading.

It looks like what I need, but I checked some reviews and it looks like a lot of people are kind of angry at it, idk why. I'll definitely consider it though.

 

[vanhees71]

Well, Dieudonne was part of Bourbaki, and they are great to collect math wisdom in a precise way. For students of math their books are, in my opinion, nightmares. You don't get any intuition from them. It's the way math is finally formulated when you found the theorems and their proof, but looking at math only in this way doesn't help to learn how to actually do mathematics. To find new theorems and proofs, I think, you need some intuition about the highly abstract "mathematical universe", but maybe I'm biased, because I'm a theoretical physicist and thus only used to the application of math to real-world problems.

 

[finnk]

Abbott's Understanding Analysis and Searcoid's Metric Spaces are, in my opinion, the clearest introduction to the subject, after that you may want to look at Pugh's Real Mathematical Analysis and Munkres's Topology, both are excellent, well written undergraduate text.

 

[AndreasC]

Abbott's Understanding Analysis and Searcoid's Metric Spaces are, in my opinion, the clearest introduction to the subject, after that you may want to look at Pugh's Real Mathematical Analysis and Munkres's Topology, both are excellent, well written undergraduate text.

Unfortunately Abbott doesn't cover the topics I am interested in. I am not looking for an intro to the basics of analysis, I have done 3 semesters of that. Neither does Pugh from what I see. The Searcoid book maybe. Please review the subjects I mentioned above!

 

[AndreasC]

Well, Dieudonne was part of Bourbaki, and they are great to collect math wisdom in a precise way. For students of math their books are, in my opinion, nightmares. You don't get any intuition from them. It's the way math is finally formulated when you found the theorems and their proof, but looking at math only in this way doesn't help to learn how to actually do mathematics. To find new theorems and proofs, I think, you need some intuition about the highly abstract "mathematical universe", but maybe I'm biased, because I'm a theoretical physicist and thus only used to the application of math to real-world problems.

Well my school is a confused hybrid of physics, pure/applied math and engineering, not pure math, so I'm probably closer to you.

Someone recommended Kolmogorov's book, do you have any opinions on that? It looks like it covers most of the things I want.

 

[vanhees71]

I don't know a book by Kolmogorov, but usually Russian books are not influenced by "Bourbakism" ;-).

 

[Adesh]

The standard textbook is Rudin's Principles of Mathematical Analysis. I like it, but it's a little terse and can be hard to read on its own. I think it's good when you're also taking a class though. It also has a good range of problems. I don't think it talks about normed vector spaces or Zorn's lemma.

I also like Carothers' book on real analysis. It covers some topics Rudin doesn't and is gentler reading in my opinion.

As I have seen and attempted some of your analysis questions (especially involving continuity, differnetiablity and functions) on Maths Challenges and was surprised to see those aspects of those topics which was very easy to miss in learning, I would like to know what were the books you personally used for Analysis? (During your student years)

 

[AndreasC]

I don't know a book by Kolmogorov, but usually Russian books are not influenced by "Bourbakism" ;-).

That's nice since it's also super cheap on Amazon apparently. At least super cheap for a textbook (why are they all so damn expensive? Good thing most are provided for free by my uni...).

 

[Infrared]

@Adesh Well, I'm still a kind of a student (graduate), but in my undergrad, I took a year-long analysis sequence that mostly followed Rudin. I also studied out of Carothers' book that I mentioned above. There are other into analysis texts that I've looked at (Tao, Pugh,...) but haven't seriously tried to learn from. I think all the analysis questions I've posed here can be solved with a good understanding of Rudin-level analysis.

I've studied a bit from big Rudin (RCA) and Folland on my own, but I don't claim to know everything in there. In particular, I don't know very much functional analysis. I've studied some other topics in analysis as needed (complex analysis in $\mathbb{C}^n$, a little elliptic PDE), but it's not really my specialty.

Anyway, I don't want to hijack this thread.

 

[AndreasC]

@Adesh Well, I'm still a kind of a student (graduate), but in my undergrad, I took a year-long analysis sequence that mostly followed Rudin. I also studied out of Carothers' book that I mentioned above. There are other into analysis texts that I've looked at (Tao, Pugh,...) but haven't seriously tried to learn from. I think all the analysis questions I've posed here can be solved with a good understanding of Rudin-level analysis.

I've studied a bit from big Rudin (RCA) and Folland on my own, but I don't claim to know everything in there. In particular, I don't know very much functional analysis. I've studied some other topics in analysis as needed (complex analysis in $\mathbb{C}^n$, a little elliptic PDE), but it's not really my specialty.

Anyway, I don't want to hijack this thread.

Do you think the other texts you mentioned or big Rudin could be closer to what I am asking?

 

[Infrared]

I admit to being a little confused by your situation. Going by the topics you listed (construction of $\mathbb{R}$, metric spaces, Arzela-Ascoli,...), your course description looks roughly at the level of the first half of (small) Rudin, but yet you've already taken three analysis courses that cover this material and need something more advanced? I think this confusion is why there's been a mismatch between the responses and your expectations.

There's no harm in looking at big Rudin (Real and Complex Analysis). It's well above the level of the topics list, but should be very appropriate for someone who has already taken 3 semesters of analysis. So take a few minutes and see if it's what you're looking for!

 

[AndreasC]

I admit to being a little confused by your situation. Going by the topics you listed (construction of $\mathbb{R}$, metric spaces, Arzela-Ascoli,...), your course description looks roughly at the level of the first half of (small) Rudin, but yet you've already taken three analysis courses that cover this material and need something more advanced? I think this confusion is why there's been a mismatch between the responses and your expectations.

There's no harm in looking at big Rudin (Real and Complex Analysis). It's well above the level of the topics list, but should be very appropriate for someone who has already taken 3 semesters of analysis. So take a few minutes and see if it's what you're looking for!

Hold on. I may have missed it while browsing through small Rudin, because when I browsed through it I found none of that.

 

[Infrared]

I'm in a bit of a hurry, so I can't do a thorough search, but: Dedekind cuts are at the end of chapter 1, metric spaces and open/closed sets are discussed in chapter 2, with many examples throughout the book. Cauchy completeness of metric spaces is in chapter 3, with the construction of the completion of a metric space given as an exercise. Countable/uncountable sets are in chapter 2. The Baire category theorem is an exercise in chapter 3. The Cantor set is discussed in chapter 2. The metric space $C[a,b]$ and relevant theorems (in particular Arzela-Ascoli) is covered in chapter 7. This covers most of the topics you listed.

 

[AndreasC]

I admit to being a little confused by your situation. Going by the topics you listed (construction of $\mathbb{R}$, metric spaces, Arzela-Ascoli,...), your course description looks roughly at the level of the first half of (small) Rudin, but yet you've already taken three analysis courses that cover this material and need something more advanced? I think this confusion is why there's been a mismatch between the responses and your expectations.

There's no harm in looking at big Rudin (Real and Complex Analysis). It's well above the level of the topics list, but should be very appropriate for someone who has already taken 3 semesters of analysis. So take a few minutes and see if it's what you're looking for!

Uh, I looked through the book again, and I got to say, well, yeah... I'm dumb... I guess I missed all that stuff due to how densely written it is. Although it still lacks some of the stuff I was looking for, like Zorn's lemma etc. But yeah, you're right, I just missed all that stuff and it just seemed like stuff I have already learned.

 

[mathwonk]

I am guessing that Kolmogorov and Fomin may be the best choice for you. I have not read it, only the table of contents, but Russian books are usually very well written and the price of the dover paperback is under $15. When the shutdown ends, you might go to a library and actually look at Dieudonne', to see what you think yourself. It is hard to read, but the content is excellent, and the problems are also superb. I am guessing K&F is much easier reading. Also K&F apparently treat Zorn's lemma, and Dieudonne' does not. But Zorn is available many places, and one should probably not choose a book based on its inclusion or not. In fact you can probably learn all you need from the wikipedia page on Zorn's lemma.

In fact used copies of the original high quality academic press hardbound book FMA by Dieudonne' are available today on abebooks for about $20.

 

[AndreasC]

I am guessing that Kolmogorov and Fomin may be the best choice for you. I have not read it, only the table of contents, but Russian books are usually very well written and the price of the dover paperback is under $15. When the shutdown ends, you might go to a library and actually look at Dieudonne', to see what you think yourself. It is hard to read, but the content is excellent, and the problems are also superb. I am guessing K&M is much easier reading. Also K&M apparently treat Zorn's lemma, and Dieudonne' does not. But Zorn is available many places, and one should probably not choose a book based on its inclusion or not. In fact you can probably learn all you need from the wikipedia page on Zorn's lemma.

In fact used copies of the original high quality academic press hardbound book FMA by Dieudonne' are available today on abebooks for about $20.

Yeah I think I'm going to get the Kolmogorov book because that will mean I can actually buy it. To be honest, usually I just download the PDFs and maybe if I feel like I should have it printed I just go to a photo copy store and I ask them to print it out for me, the price is much lower than buying the books and I just can't afford them. But this is pretty affordable so I'll probably buy it. I also noticed Amazon offers a "package deal" with 2 other books by him which also seem very interesting (one is on functional analysis which I need, the other is probability theory which I may or may not immediately need but it's good to have I guess because the textbook I was given from my uni is not great). Only issue is the shipping is kinda expensive...

Although I read some reviews saying the book had mistakes which makes me a bit skeptical...

 

[AndreasC]

You might take a look at Dieudonne's Foundations of Modern Analysis. It's on a rather high level and is demanding but rewarding reading.

Alright, time for an update, a few months later, I've read through Kolmo, and I am now reading through Dieudonne. Credit to mathwonk, as I now wish I had started with Dieudonne. Not that Kolmo was bad. But 1) it did have many typos etc which made it a bit hard to go through sometimes but more importantly, 2) it was very encyclopedic and better suited as a reference. Like, it is absolutely filled with info and theorems about many different things presented in a somewhat Wikipedia-like format, without much to tell you that, hey, this is important, pay attention etc (this caused me to skip a couple of important things that I didn't judge at the time to be important, only to find out 50 pages later that, well, I should have paid attention). As I said, that's great as a reference but it was somewhat grating to go through. I don't see what people are talking about Dieudonne being too hard or abstract or whatever. I think it's fine and nicely organized (the order of presentation is also somewhat similar to Kolmogorov and different to, say, Rudin, which I browsed a bit and found somewhat confusingly organized). But that may be because I already have some knowledge on the subject now.

 

[mathwonk]

Thank you for the feedback. Since you appreciate Dieudonne', I recommend also a much more traditional book as a complement to it, namely vol. 2 (also vol.1), of Courant's Differential and Integral Calculus. When Dieudonne' says in his preface "It is clear students should have a good working knowledge of classical analysis before approaching this course", one fulfills this prerequisite with Courant.

Another classical analysis work is one by Dieudonne called Infinitesimal Calculus, but it is hard to find. If it is the same book I perused once, the preface states that a common complaint (in about 1970) was that "les etudiants d'aujourdhui ne savent pas calculus". I.e. "students of today no longer know how to calculate", and he proceeds to try to remedy this problem. I have not read it in detail, but I was impressed with the little part I saw.

 

[vanhees71]

Well, but the Bourbaki school precisely make students of pure mathematics not to appreciate calculational abilities anymore. Bourbakism is great as a formal program how to unify math in its formal aspects but it kills the intuitive part, which I think (though I'm a theoretical physicist rather than a mathematician, so I might be wrong) is important in math too to find new theorems and proofs etc.

 

[AndreasC]

it kills the intuitive part

I don't know if that is entirely true. In the book I am reading by Dieudonne for example, I feel like he did a good job of emphasising the geometric intuitions for, say, metric spaces. In contrast, Kolmogorov and, from what I've seen, Rudin too, I'd say did less to emphasise it. On the other hand, what Kolmogorov had which Dieudonne lacks is many concrete examples.

 

[mathwonk]

I also think Bourbaki gets a bad rap that to me at least is not borne out by reading Bourbaki books. In the ones I have been consulting, not only is the theory explained well, but the problems seem excellent and there is even a very rare, perhaps unique, discussion of the history of the subject as well. I admit I skip over some of the very formal foundational volumes (algebraic structures) that do not appeal to me and go straight to those that treat my topics: like linear algebra, or primary decomposition and flatness. When I do this I usually find important aspects of the subject covered well that are omitted in other books.

I admit also I was myself surprized to read that complaint about lack of computation from Dieudonne when his own books are so abstract. But then I realize that he apparently was indeed assuming that a student should take his advice seriously, and not even dream of approaching his abstract treatement without first mastering a classical treatment of analysis, whereas in fact students at places like Harvard in my day were being thrown headfirst into the abstract version completely unprepared. Indeed I may be wrong, but as I recall, in the 1960's, if one took the traditional several variables calculus class Math 105, one could not then also take the abstract Dieudonne type course, Math 55, for credit! Apparently today one can also not combine 55 with the Intro to analysis math 112. I solved this in my day by taking 55 for credit and just auditing 112.