Anyone know a good, rigorous introductory analysis text?
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".
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.
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.
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.
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. ]
How's Apostol's Calculus compared to Spivak's? Are they essentially the same thing?
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.
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.
Alright, I'm gonna 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?
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.)
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...
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??
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!
Oh I thought analysis and real analysis are two different stuff
I know Spivak's gonna 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...
Well, in this case they were. Real analysis is more specific, to distinguish it from other branches, like functional analysis, numerical analysis, etc.
They're both excellent, as is Apostol's Mathematical Analysis. But after Spivak, you might be able to decide this for yourself better.
So in short, I can't go wrong with any of them?
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