Looking for a Comprehensive Introductory Analysis Text?

In summary: Spivak's lectures are really dense and you don't need the new material. That's why I generally recommend people to go straight to Pugh's book after finishing Spivak.In summary, Spivak's Calculus is a good book for those who have never seen theoretical calculus before, while Apostol's Calculus is good for those who already know theoretical calculus. Both books have the same information, but Apostol's is more beginner-friendly.
  • #1
thrill3rnit3
Gold Member
716
1
Anyone know a good, rigorous introductory analysis text?
 
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  • #2
How introductory do you want?

If you've never seen theoretical calculus, see Spivaks Calculus or Ross' Elementary Analysis. If you already know that the best choice is Pugh "Real Analysis".
 
  • #3
I think Rudin's "Principle of Mathematical Analysis" is good if you're up for the challenge, but it might be more helpful to read Spivak's "Calculus" first to get a taste of some analysis.
 
  • #4
Take a look at Simmons' "introduction to Topology and Modern Analysis". It includes operators, Hilbert Space, Banach Spaces and Algebras and much more in a rigorous manner.
 
  • #5
Howers said:
How introductory do you want?

If you've never seen theoretical calculus, see Spivaks Calculus or Ross' Elementary Analysis. If you already know that the best choice is Pugh "Real Analysis".

aostraff said:
I think Rudin's "Principle of Mathematical Analysis" is good if you're up for the challenge, but it might be more helpful to read Spivak's "Calculus" first to get a taste of some analysis.

alright, I'll go ahead and check out Spivak's calculus first.

I'm a high school soph and I just finished elementary diffy-q's, so it's safe to say I've never seen theoretical calculus before. I mean I think I have, but I just haven't actually read it yet.
 
  • #6
I'm a high school soph and I just finished elementary diffy-q's, so it's safe to say I've never seen theoretical calculus before.
Then Spivak's Calculus or Apostol's Calculus (yeah, original titles) is definitely the best choice in my opinion.

[After these introductions, there are a lot of good real analysis textbooks. Of course Rudin is well-known, but terse. I like Apostol's Mathematical Analysis. Abbotts' Understanding Analysis, Pugh's Real Analysis, Dieudonné's Foundations of Modern Analysis, and others. ]
 
  • #7
How's Apostol's Calculus compared to Spivak's? Are they essentially the same thing?
 
  • #8
As always I recommend Marsden and Hoffman's Elementary Classical Analysis. It is at the same time beginner-friendly and treats things in full generality.
 
  • #9
thrill3rnit3 said:
How's Apostol's Calculus compared to Spivak's? Are they essentially the same thing?

Apostol mostly shows important results, which I like very much. Spivak makes you derive a lot of important thing, but his reading is very fun until you hit extremely difficult problems. In the end, both books contain the same information but if you don't like finding answers yourself you might prefer Apostol (you can still do this in Apostol, just cover the proof and try and work it out yourself). Also, Apostol has a volume 2 covering multivariable calulus. Spivak sort of does this with his manifolds book, but I find his treatment way too terse. Lastly, I think Spivak is overkill - good or bad depending on whether you have the time.
 
  • #10
Alright, I'm going to reserve Spivak's calculus from the public library, as it seems like that's what most of you guys are recommending.Lookss like they only have the edition from 1967...do you think that's good enough?
 
  • #11
I echo the recommendation for Spivak's book, and yes, the 1967 edition is fine (it's what I have). The later editions added a few chapters covering a bit more material, but the 1967 has all the key stuff on limits and continuity, sequences and series, differentiation and integration.

This is one of the best math books in existence, in my opinion. It's extremely well written, it doesn't presume you know any of the material already but assumes the reader is intelligent and capable of learning and applying calculus with full mathematical rigor.

The exercises are fantastic: none of them are rote monkey-see-monkey-do drill problems as in most calculus books; many require proofs and all demand careful thinking; some of them are very, very challenging even if you have mastered all the theorems and are deft with your epsilons and deltas.

If you learn calculus from Spivak, including doing a lot of the exercises, then you'll be very well prepared for Rudin's "Principles of Mathematical Analysis," which is also a very beautiful book and for my money the one truly indispensable reference for this material, but one which I think can only really be appreciated once you have learned basic analysis/rigorous calculus elsewhere first (or if you are using it in an academic course where you have a teacher to supply the motivation and help you digest Rudin's austere exposition and proofs.)
 
  • #12
alright so I'll probably stick with the 1967 edition for now and then I'll just get the most recent edition in the future...
 
  • #13
Don't waste your money and time. Just move on to a real analysis book after Spivak. I recommend Pugh's "Real Mathematical Analysis".
 
  • #14
qspeechc said:
Don't waste your money and time. Just move on to a real analysis book after Spivak. I recommend Pugh's "Real Mathematical Analysis".

Why should I jump straight to real analysis? Shouldn't I read an analysis book first? Say, Rudin's??
 
  • #15
Analysis = real analysis = Rudin = Pugh :)

qspeechc meant that after you finished Spivak, don't bother to buy the new edition or put more money/effort in it, but go to (real) analysis. Indeed, Rudin or Pugh are excellent choices after Spivak. But first things first, Spivak will keep you busy for a while!
 
  • #16
Oh I thought analysis and real analysis are two different stuff :rolleyes:

I know Spivak's going to make me busy during the summer, and this is probably a question a bit early to ask, but which one would you recommend, Pugh or Rudin? So I know what to look for right after I finish reading...
 
  • #17
thrill3rnit3 said:
Oh I thought analysis and real analysis are two different stuff :rolleyes:
Well, in this case they were. Real analysis is more specific, to distinguish it from other branches, like functional analysis, numerical analysis, etc.
I know Spivak's going to make me busy during the summer, and this is probably a question a bit early to ask, but which one would you recommend, Pugh or Rudin? So I know what to look for right after I finish reading...
They're both excellent, as is Apostol's Mathematical Analysis. But after Spivak, you might be able to decide this for yourself better.
 
  • #18
So in short, I can't go wrong with any of them?
 
  • #19
Correct.
 

FAQ: Looking for a Comprehensive Introductory Analysis Text?

What is "Introductory Analysis Text"?

"Introductory Analysis Text" is a textbook designed for students who are learning the basics of analysis in mathematics. It covers various topics such as limits, derivatives, and integrals.

Who is the target audience for this textbook?

The target audience for this textbook is typically undergraduate students who are majoring in mathematics or a related field. It can also be useful for high school students who are preparing for college-level math courses.

What makes "Introductory Analysis Text" different from other analysis textbooks?

One of the main differences is that "Introductory Analysis Text" focuses on providing a clear and intuitive understanding of the concepts, rather than just memorizing formulas and procedures. It also includes many real-life examples and applications to help students see the practical relevance of the material.

Is any prior knowledge of calculus required to understand this textbook?

Yes, some prior knowledge of calculus is necessary for understanding this textbook. Students should have a good grasp of basic calculus concepts such as limits, derivatives, and integrals before diving into analysis. Some textbooks may also assume knowledge of pre-calculus topics like functions and graphs.

How can "Introductory Analysis Text" be used in a classroom setting?

This textbook can be used as a primary resource for a traditional lecture-based course or as a supplement to other materials in a flipped classroom setting. It also includes exercises and problems for students to practice and apply their understanding of the material.

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