Introduction to Analysis: Serge Lang vs Pugh

[ronaldor9]

Which book should I use as a introduction to analysis:
Serge Lang undergraduate analysis
Pugh Real Analysis

I've looked through both books and both seem great however maybe Pugh is a little too advanced? I haven't studied analysis before so I'm not sure if i should go through Serge's first then Pugh or just to finish one of the books then move on to other subjects instead of wasting time reading another book on analysis when I could be studying more advanced things.

 

[Landau]

Certainly, one of the two is enough, and they are at about the same level. Lang covers a lot of material, more than a you can do in a two-semester course. They're both great, though. I think you can't go wrong with either.

 

[Pinu7]

Out of the two, I would pick Serge Lang's book.

But also consider Elementary Classical Analysis by Marsden or Mathematical Analysis by apostol.

 

[n!kofeyn]

I remember browsing Pugh's book and found it to be unique and interesting. He covers some neat material in chapters 4 and 5 that I wasn't exposed to in my undergraduate studies. I've never read Lang's book, but I wasn't impressed by his complex analysis book. I've listed some other choices below as well.

If you are just starting out, particularly with proofs, then I would recommend:
https://www.amazon.com/dp/0131481010/?tag=pfamazon01-20 by Steven Lay
It has good discussions on techniques of proofs, logic, etc., and then has a very readable and great introduction to analysis.

Also, https://www.amazon.com/dp/038790459X/?tag=pfamazon01-20 by Kenneth Ross has been highly recommend before as a good introduction, and the reviews on Amazon agree.

 

[naele]

The book by Lang is absolutely awful, plain terrible. Its only redeeming feature is the companion solutions manual. I would recommend David Brannan's A First Course in Mathematical Analysis or the above mentioned book by Steven Lay.

 

[Klockan3]

What's wrong with Rudin?

 

[mosenja]

What's wrong with Foundations of Modern Analysis by Dieudonné?

 

[Landau]

The book by Lang is absolutely awful, plain terrible.

Why?

What's wrong with Rudin?

It's a bit too terse for my taste. Nice for reference, not ideal as introduction.

What's wrong with Foundations of Modern Analysis by Dieudonné?

Not much, but it is also very terse. I don't think I would have liked to learn analysis for the first time out of Dieudonné. Now I'm doing functional analysis there's a lot of useful stuff in it.

 

[Klockan3]

It's a bit too terse for my taste. Nice for reference, not ideal as introduction.

I dunno, I liked that quality of the book. Much better than the books with a lot of fluff everywhere and the topics it brings up aren't that advanced till you get to the last chapters.

But maybe I were too damaged by maths already to see it, I took a few for fun analysis courses without any literature or such before the one with Rudin, it was like "zomg a book! lol ez".

 

[ronaldor9]

The book by Lang is absolutely awful, plain terrible. Its only redeeming feature is the companion solutions manual. I would recommend David Brannan's A First Course in Mathematical Analysis or the above mentioned book by Steven Lay.

Why? I used his introductory first year calculus book and found that it was excellent, I concurrently also used another book as additional reference and found Lang's was much better at explaining theory. But I've heard that some people simply don't like his writing style.

 

[naele]

It's not so much his writing style. I flipped through about 150 pages and found only one worked example. That's mostly why I didn't like it.

 

[Landau]

It's a book about analysis, not calculus ;) There are a lot of worked examples: every theorem with proof is one!

 

[Pinu7]

Landau is right, math books do not even need proofs! If an author wanted to, he could make a book with just definitions and theorems without proof, illustration, or further explanation. It might actually be better that way.

 

[Landau]

I don't agree. Of course we need proofs, explanations, illustrations, etc. But 'worked examples', with the emphasis on 'worked', are things you will find in calculus books. E.g. 'calculate a normal vector to this given plane', things you can solve with a routine algorithm. The exercises in an analysis course mainly consist of proving statements. I don't know what a 'worked example' is in this context, besides the proofs of theorems.

 

[naele]

By worked example I meant, for example, proving that a sequence converges to a limit using the epsilon-N definition. Or, proving a sequence is Cauchy, epsilon-delta proofs of limits of functions, continuity at a point, uniformly continuous, etc. I think these exercises are important because they reinforce the definitions. Imagine trying to prove something without fully understanding what you're working with.