Seeking books that cover my upcoming math course

[bacte2013]

Hello!

I am currently searching for some alternative books I can use for the analysis course starting on this Fall Semester. The course will cover the compactness, contraction principles, approximation theory, and some applications like special functions and Fourier series. The required textbook is Rudin-PMA, but I do not like that book. Could you recommend some alternative books that cover those topics?

The sample syllabus of the course (MATH 522) is here:
http://www.math.wisc.edu/sites/default/files/521-522_0_1.pdf

 

[smodak]

This is not an Analysis Book but I think this will be a very important book for you to read before taking the course.
How to Think About Analysis Alcock
https://www.amazon.com/gp/product/0198723539/?tag=pfamazon01-20

I suggested some books that I like and that cover most of your topics here

Hope this helps.

 

[mathwonk]

lang's undergraduate analysis, ALSO LISTED ON YOUR SYLLABUS, IS MUCH BETTER THAN RUDIN. TRY THAT. oops, darn caps lock.

here's one for $10:

http://www.abebooks.com/servlet/SearchResults?an=lang&sts=t&tn=undergraduate+analysis

 

[bacte2013]

lang's undergraduate analysis, ALSO LISTED ON YOUR SYLLABUS, IS MUCH BETTER THAN RUDIN. TRY THAT. oops, darn caps lock.

here's one for $10:

http://www.abebooks.com/servlet/SearchResults?an=lang&sts=t&tn=undergraduate+analysis

I meant MATH 522, which will cover topics like compactness, approximation theory, differential forms, Stoke's theorem, etc. The Math 521, which I already took, covers standard topics in Rudin and Lang (by the way, I did not like Lang's Undergraduate Analysis et al.)...my favorites are Hairer/Wanner and Rudin.

 

[micromass]

I meant MATH 522, which will cover topics like compactness, approximation theory, differential forms, Stoke's theorem, etc. The Math 521, which I already took, covers standard topics in Rudin and Lang (by the way, I did not like Lang's Undergraduate Analysis et al.)...my favorites are Hairer/Wanner and Rudin.

The name is Stokes, not Stoke. So it's Stokes' Theorem. Very common error.

 

[mathwonk]

I am a little puzzled as to my knowledge Lang's Undergraduate analysis does cover all those topics, compactness very early, and stokes very late, including the differential forms case. my old copy of the book is from its inception, when it was titled Analysis I, but the table of contents I viewed of UA apparently had the same topics. But if you prefer Rudin to Lang, I think I should not advise you since our tastes are so different. yes here is a link, compactness on page 193, differential forms and stokes on page 607:

https://www.amazon.com/dp/0387948414/?tag=pfamazon01-20

and as spivak points out, the theorem, although published by stokes, seems due to lord kelvin. i thank you for pointing out the very scholarly looking book by hairer and wanner, which was unknown to me. some people might call rudin scholarly as well, but to me "tedious" is more descriptive. whereas rudin seems never to give insight, hairer and wanner do, e.g. when before their proof of existence of the riemann integral of a continuous function, they observe that the key point is uniform continuity.

 

[bacte2013]

The name is Stokes, not Stoke. So it's Stokes' Theorem. Very common error.

I see...Also, sometimes I noticed that some books addressed names of mathematicians different from standard. For example, Engelking's Topology: A Geometric Appaora

I am a little puzzled as to my knowledge Lang's Undergraduate analysis does cover all those topics, compactness very early, and stokes very late, including the differential forms case. my old copy of the book is from its inception, when it was titled Analysis I, but the table of contents I viewed of UA apparently had the same topics. But if you prefer Rudin to Lang, I think I should not advise you since our tastes are so different. yes here is a link, compactness on page 193, differential forms and stokes on page 607:

https://www.amazon.com/dp/0387948414/?tag=pfamazon01-20

and as spivak points out, the theorem, although published by stokes, seems due to lord kelvin. i thank you for pointing out the very scholarly looking book by hairer and wanner, which was unknown to me. some people might call rudin scholarly as well, but to me "tedious" is more descriptive. whereas rudin seems never to give insight, hairer and wanner do, e.g. when before their proof of existence of the riemann integral of a continuous function, they observe that the key point is uniform continuity.

It is fine. I actually find Barbarian's Fundamentals of Real Analysis + Kolmogorov/Fomin books to cover what I need. They go beyond Lang and Rudin in terms of compactness, approximation theory, etc.

Personally, I think Hairer/Wanner can be a great book to start learning analysis, compared to contemporary books like Rudin, Lang, Strichartz, so on. It is a gem that describes details like approximation of numbers that I did not learn from other books. Perhaps my opinion is based as I found historical approach to be very entertaining.

 

[mathwonk]

yes i like berberian's book as well, but it is essentilly a book on measure and integration, not calculus of normed spaces as lang's is. it does not cover differential forms or stokes. it does cover compactness in metric spaces, characterizing it as equivalent to "complete and totally bounded", a theorem i also enjoyed as a student. but lang proves all the basic results in his setting, equating the henie borel, bolzano weierstrass, and closedness and boundeness properties. i agree that berberian writes in a more careful user friendly way than lang, as perhaps do hairer and wanner, and kolmogorov fomin, but i do not recommend rudin as at all in this company.

 

[bacte2013]

yes i like berberian's book as well, but it is essentilly a book on measure and integration, not calculus of normed spaces as lang's is. it does not cover differential forms or stokes. it does cover compactness in metric spaces, characterizing it as equivalent to "complete and totally bounded", a theorem i also enjoyed as a student. but lang proves all the basic results in his setting, equating the henie borel, bolzano weierstrass, and closedness and boundeness properties. i agree that berberian writes in a more careful user friendly way than lang, as perhaps do hairer and wanner, and kolmogorov fomin, but i do not recommend rudin as at all in this company.

I found that Lang also wrote "Real and Functional Analysis". It seems to cover concepts like Banach space and differential forms too, and I am curious if that book builds upon his undergraduate analysis. If I buy Lang's Undergraduate Analysis, can I assume that I can skip the first section and start from the second section? It seems that the first section is basically the advanced calculus, which is more than sufficiently covered by H/W.

I do agree that Rudin is not a good textbook (after reading that book, all I remembered was a set of definition, theorems and proofs that I could not make a connection). I actually got an opportunity to do a reading course in the complex analysis, and I first thought about Rudin-RCA, but I chose Barry Simmons' two-volume set as his books emphasize topology and covers interesting topics from number theory. That is good as I love topology so much (I still read and read again topology books by Singer/Thorpe and Engelking).

 

[mathwonk]

Lang's two books were originally titled Analysis I and Analysis II, so he did mean the second one as a continuation of the first. I like the first better, and find the second one very abstract and difficult. Still his treatment of measure theory and integration there, although abstract, is insightful. One thing I dislike is his insistence on using Banach space valued functions, as Dieudonne' did. This just makes it harder to learn, and does not add anything. True, the arguments are almost the same, but in that situation I recommend learning the elementary version and then just saying that, for those who know about Banach spaces, the generalization is the same, rather than encumbering the first encounter with the full generality.

I myself would not skip any of Lang's first volume, since he may well make some things you may already know seem even more clear and simple than other books do. I myself like to keep relearning and restudying things until they reduce in my mind to elementary principles.