Analysis textbook recommendation

[ELB27]

Hi everyone,

I just finished self-studying linear algebra from Treil's Linear Algebra Done Wrong, and I want to continue with analysis.
I don't know the requirements, so I'll just list what I know. I already know single and multi-variable calculus (to give a sense of my level, I knew everything necessary math-wise to comfortably go through Griffiths' Introduction to Electrodynamics), linear algebra (Treil's book is my only exposure but I went through it from cover to cover and did all the problems) and a bit of differential equations (only the very basic stuff - separation of variables of ODEs and PDEs and integrating factors).

I want to self-study analysis because I'm interested in a rigorous approach to calculus and because I liked the proof-writing and the more abstract (for me) approach of Treil's book. However, I do not know where to begin with it and which book(s) would be suitable for my level/interest.

Thanks in advance for any suggestions and explanations!

 

[Geofleur]

Stein and Shakarchi's Princeton Lectures in Analysis series is very good. It's four books though, so you'd need to have the time to relax and settle into it! There's also a real analysis book by Royden (note that I did not say Rudin - Rudin is good but perhaps not for starting out!). Royden's book has nice, understandable proofs.

 

[ELB27]

Stein and Shakarchi's Princeton Lectures in Analysis series is very good. It's four books though, so you'd need to have the time to relax and settle into it! There's also a real analysis book by Royden (note that I did not say Rudin - Rudin is good but perhaps not for starting out!). Royden's book has nice, understandable proofs.

Thanks! I'll have a look. Concerning the Princeton Lectures, these seem to be 4 books on 4 different areas of analysis. In what order should I tackle them should I choose this series?

 

[Geofleur]

I would read them in order, though the 3rd volume is where most of the usual topics in real analysis are. The 1st volume is on Fourier analysis and the 2nd on complex analysis. The 4rth volume is functional analysis.

 

[Ahmad Kishki]

I almost had the same background when i started Real Analysis from Stephen Abbott's Understanding Analysis, i would definitely recommend it for the historical background, clarity of proofs, and quality of exercises. However, i would advise you to learn some (basic) set theory and logic before going into analysis - that would be a major asset in proof writing and reading.

 

[ELB27]

i would advise you to learn some (basic) set theory and logic before going into analysis - that would be a major asset in proof writing and reading.

Thank you for the suggestion! I will have a look.

I picked up some set theory here and there (mainly the notations). Do you think I should devote some time exclusively to it? If so, can you recommend a good resource?

 

[SrVishi]

Just going to jump in here. I personally think set Theory is completely worth the time. Now, you don't have to delve too deep into it. You could use Halmos' Naive Set Theory to get everything you'll ever use in analysis. My favorite set Theory book personally is Introduction to Set Theory by Jech and Hrbacek. It may be overkill for what you want, but it is worth seeing at least some point in your mathematical career. Now, outside of the beauty and foundational prowess of set Theory in mathematics, seeing set Theory in a single, focused book, means you can really skip the set Theory review chapters in basically every book in other subjects, whose set theoretic chapters are often horribly rushed. For your level, I would again say take a look at Naive Set Theory.

 

[Anama Skout]

I really loved Terence Tao's Analysis I & II (didn't finish them, just read some chapters), his writing is crystal clear. However I'm not sure if he covers all the topics you'll need and there's a lack of geometric pictures.

Edit: Another very interesting text is Zorich's Mathematical Analysis I & II. I'm told that it has a lot of physical examples, and it is - of course - much more complete than Tao's book, here's what it covers:

EDIT: To have a taste of what the book looks like you may check these two links http://math.univ-lyon1.fr/~okra/2011-MathIV/Zorich1.pdf & http://math.univ-lyon1.fr/~okra/2011-MathIV/Zorich2.pdf from the university of Lyon, which provides a free copy of the two volumes.

 

[Ahmad Kishki]

Thank you for the suggestion! I will have a look.

I picked up some set theory here and there (mainly the notations). Do you think I should devote some time exclusively to it? If so, can you recommend a good resource?

The first three chapters of How to Prove It, by Velleman are sufficient for a first course on real analysis, however more is better.

 

[Hercuflea]

I think any student of analysis should read this book:

https://www.amazon.com/gp/product/1107694043/?tag=pfamazon01-20

(At least read the first 4 chapters before you begin studying analysis, Ideally read the whole thing.)

 

[ELB27]

Thank you all for your suggestions! I went through all the books and I intend to go over Halmos' Naive Set Theory and then move to Zorich's books which I find very impressive in terms of breadth and depth. If I will struggle, I will take more time to go through Abbott's text which, from what I gather, is a very good introduction to the mindset of Analysis.

One last question regarding Zorich's text: the first chapter deals with logical operations with which I'm not very familiar. Is there anything more to them than their true/false table definitions (when A is true/false and B is true/false, some logical of A and B is true/false)?
(For instance, the very first exercise asks me to prove that: $\neg(A∧B) ⇔\neg A∨\neg B$. Is there more to it than writing the true/false table of both the RHS and LHS and noting that they are the same?) If there is indeed more to it, what would be a good resource to study it from in more depth?

 

[Ahmad Kishki]

Thank you all for your suggestions! I went through all the books and I intend to go over Halmos' Naive Set Theory and then move to Zorich's books which I find very impressive in terms of breadth and depth. If I will struggle, I will take more time to go through Abbott's text which, from what I gather, is a very good introduction to the mindset of Analysis.

One last question regarding Zorich's text: the first chapter deals with logical operations with which I'm not very familiar. Is there anything more to them than their true/false table definitions (when A is true/false and B is true/false, some logical of A and B is true/false)?
(For instance, the very first exercise asks me to prove that: $\neg(A∧B) ⇔\neg A∨\neg B$. Is there more to it than writing the true/false table of both the RHS and LHS and noting that they are the same?) If there is indeed more to it, what would be a good resource to study it from in more depth?

The equation follows directly from deMorgan's Laws. I bet you will find this in most set theory books :)

 

[Anama Skout]

Is there more to it than writing the true/false table of both the RHS and LHS and noting that they are the same?)

For that exercise, no, just write the truth table of both the LHS & RHS and conclude.

However I see that Zorich doesn't introduce proof techniques (e.g. how to prove a biconditional? how to make a proof by contradiction? how to make a proof by contrapositive? ...) Are you already familiar with them? Because you'll absolutely need them in any analysis text

 

[SrVishi]

Does that mean you aren't that familiar with formal logic and deduction rules? If that is the case, I would suggest reading Book of Proof (freely available online). It covers this stuff a but, at least enough to get you through Zorich's book. It contains some examples from set Theory, and other parts of mathematics that would be a fantastic exposure for higher level math. Another good one is How to Prove it by Velleman, but I'm not so sure how it conpares in quality to the previous one mentioned. Just remember to not rush anything. If you rush, you lose understanding, pateince, and most important of all, precious interest and intrinsic motivation. Take your time!

 

[Anama Skout]

You can download the entire Book of Proof here: http://www.people.vcu.edu/~rhammack/BookOfProof/BookOfProof.pdf

Or if you want individual chapters you may have a look here: http://www.people.vcu.edu/~rhammack/BookOfProof/

Part I & II & III are sufficient. Zorich already treats functions and relations and cardinality.

 

[ELB27]

However I see that Zorich doesn't introduce proof techniques (e.g. how to prove a biconditional? how to make a proof by contradiction? how to make a proof by contrapositive? ...) Are you already familiar with them? Because you'll absolutely need them in any analysis text

Yes, I am familiar with these techniques but I never learned them in a specialized setting (learned them on the fly during self-study of linear algebra from LADW).

Does that mean you aren't that familiar with formal logic and deduction rules?

Not sure what you mean by those "rules" but I have done some (perhaps semi-)formal proofs, again from LADW.

You can download the entire Book of Proof here: http://www.people.vcu.edu/~rhammack/BookOfProof/BookOfProof.pdf

Or if you want individual chapters you may have a look here: http://www.people.vcu.edu/~rhammack/BookOfProof/

Part I & II & III are sufficient. Zorich already treats functions and relations and cardinality.

Thank you very much! Judging by the contents, this book seems perfectly suited to complete my knowledge of the various techniques.
(I am not trying to save time as much as to save money on books. However, I prefer to read the lengthy/heavy texts from a printed book rather than from a screen. That's why I would like to buy Zorich's text but study the requisites online.)

 

[SrVishi]

I mean rules of inference (deduction) and the ideas of logical equivalence. The proof methods involving how to prove a biconditional that you are apparently familiar with is of course an example. You could perhaps find the techniques you've been exposed to to be sufficient, but some more technical details may go over your head if you don't know formal logic, which is discussed in proof book like book of Proof. For example, do you know how to integrate the rule of distribution or DeMorgan's laws when present in a chain of conjunction with disjunctions of statements? This is important in proving that closed subsets of a metric space are closed under arbitrary intersections, by using DeMorgan's laws and the fact that a set is open if and only if it's complement closed. Also, do you know that a statement "p or q" is equivalent to "if not p, then q"?, because that is an extremely convenient way of proving a disjunction statement. Also, truth tables are good to know since you can learn how to turn a proof problem you are faced with, using logical equivalences, into proving a statement that you know how to work with. If this seems unfamiliar to you, then you really should focus more first on logic. It's also a really fun subject!

 

[ELB27]

I mean rules of inference (deduction) and the ideas of logical equivalence. The proof methods involving how to prove a biconditional that you are apparently familiar with is of course an example. You could perhaps find the techniques you've been exposed to to be sufficient, but some more technical details may go over your head if you don't know formal logic, which is discussed in proof book like book of Proof. For example, do you know how to integrate the rule of distribution or DeMorgan's laws when present in a chain of conjunction with disjunctions of statements? This is important in proving that closed subsets of a metric space are closed under arbitrary intersections, by using DeMorgan's laws and the fact that a set is open if and only if it's complement closed. Also, do you know that a statement "p or q" is equivalent to "if not p, then q"?, because that is an extremely convenient way of proving a disjunction statement. Also, truth tables are good to know since you can learn how to turn a proof problem you are faced with, using logical equivalences, into proving a statement that you know how to work with. If this seems unfamiliar to you, then you really should focus more first on logic. It's also a really fun subject!

Indeed, these things do go over my head, I am barely familiar with DeMorgan's law in its formal form - I will take the time to learn them. I already started going through the book of proof linked above - does it cover these subjects? It does cover logic in Section 2, is the level there sufficient?

 

[SrVishi]

Yes, book of proof does contain the sufficient logic background. There are a tiny few places where I would personally explain something in a different way than he did, but he does all that he needs to, and in a good manner. I just found the book "How to Prove It" by Velleman online here: https://opeconomica.files.wordpress.com/2014/08/daniel-j-velleman-how-to-prove-it.pdf it is what I used for learning proofs, and it is really good. If you happen to end up really liking logic and want to learn it further, here is a free ebook logic textbook (though I haven't read it, it should be good, but if you want, I could talk to you about the book I personally used) http://rocket.csusb.edu/~troy/SLmain.pdf
Now that you have access to both, I think it is important to discuss what the main differences between Book of Proof and How to Prove it are (and anyone else could expand upon this). First of all, How to Prove it is slightly more in-depth for logic and all, though I don't favor Velleman's way of explaining inference rules in a seprarate manner. For example, he talks about Modus Ponens, an inference rule, many pages away from the other inference rules, instead of unifying all of them together for the beneft of the reader, but that may just be nitpicking. Velleman does, on the other hand, give at least an incredibly small idea about the notion of column proofs, which I think would be nice to at least see as it would help the reader conceptualize such an organized manner of reasoning, but again, it is not entirely necessary. If you did want to see it, you could easily look u column proofs in other resources like that logic textbook. I also believe Velleman has more exercises, which exercises are really good, some of his exercises kind of become trivial repeats, so I won't exactly put much favor on Velleman's book in this case. Now, for the biggest difference between the two, they use different examples. Alot of proof books teach proof methods and logic and then proceed with examples and exercises from various forms of mathematics. These two books differ from where they take those examples. Book of Proof is kind of a hand-bag where it takes things from various fields, and includes some set theory. Velleman, on the other hand, focuses more exclusively on set theory, and actually goes fairly in depth for a proof-book (note: this is naive set theory, not axiomatic). Overall, Either book will leave you at a great place, and definitely feel free to read one, and refer to the other for either an alternative explanation, or just extra material. This is what mathematicians an math students generally do anyways: read multiple books at one (again, usually by way of having a "main" text, and then one or more of supplementary or "secondary" books). Zorich's book afterwards seems like a good idea for you after these books, and do read through the super short section on proofs in the book just for review. He seems to cover some topic that can be more used for application, since that is what I assume you really want to use it for, correct?

 

[ELB27]

Overall, Either book will leave you at a great place, and definitely feel free to read one, and refer to the other for either an alternative explanation, or just extra material. This is what mathematicians an math students generally do anyways: read multiple books at one (again, usually by way of having a "main" text, and then one or more of supplementary or "secondary" books). Zorich's book afterwards seems like a good idea for you after these books, and do read through the super short section on proofs in the book just for review. He seems to cover some topic that can be more used for application, since that is what I assume you really want to use it for, correct?

Thank you very much for your detailed answer! I really appreciate it!
I will definitely refer to Velleman as you suggested (and kindly provided a link to). I don't have a particular application in mind for Zorich. Ideally I want a rigorous and comprehensive treatment of analysis with a slight tendency towards applications and his book seems to achieve just that.

Thanks again! I will bookmark this thread to keep your comments in mind when I get to the respective parts of the book(s) you wrote about.