Modern Analysis Preparation

Main Question or Discussion Point

I will be taking Modern Analysis at Columbia University this summer and would like some suggestions on being as prepared as possible.

I have taken:
• Non-rigorous Calc I, II, III
• Intro to Probability & Statistics
• Intro to Linear Algebra
• Discrete Math I, II
• Various Computer Science courses
• Differential Equations, and a
• Proof based "Transition to Advanced Math Course" (it covers and introduction to: logic, counting, relationships, and functions.)

I have limited time between the end of the semester and the beginning of the summer session. The following is a list of preparation topics that were recommended on other posts.

Could you please prioritize the list from most important to review to least important to review. In addition, please provide your knowledge of the course content (e.g. professor, math grad student, student taking Analysis, student that has taken Analysis):

Review:
1. the course text, by Rudin
2. review Calc texts by Spivak / Apostall (clarification on specific topics would be helpful)
3. review Analysis books by Spivak / Apostall (clarification on specific topics would be helpful)
4. review series & sequences
5. review delta epsilon proofs
6. practice proof writing (recommendations on a text with basic proof writing problems and, more importantly, lots of detailed solutions would be appreciated.)

Thank you in advance for you help.
Bizoid

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lurflurf
Homework Helper
The most useful thing to do is to understand the delta-epsilon formulation of limits
See if you can show some limits exist or have some value
next get a list of calculus topics and see if you understand them all and can do computations
when you can pick a few and either try to understand the proofs in some nice book or provide your own.
unless you have alot of time you will not be able to revise all topics since there are many, but some will helt alot.
Another good thing to do is think about when a method can be used and why it cannot be used other times. Often upon first exposure one misses some details.
And more abstraction see if your theorems
for f:R->R
hold if f:E->F with E and F banach spaces
which things for continuous function hold or are slightly changes if f is piecewise continous or monotone

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mathwonk
Homework Helper
i presume you mean the course text is "baby rudin", principles of analysis.

this book is very terse and hard to read and learn from. much better is spivak's calculus book, which covers much of the same material.

you might start reading rudin and use spivak to supplement it.

the treatment of real numbers from rudin, a la dedekind cuts, is sheer masochism.

the point is to prove rigorously, that if you have the rationals, you can consider reals to be gaps between intervals of rationals, and even define and prove, all the arithmetic operations, on these intervals of rationals.

this is rigorous practice at manipulating inequalities, but has no actual mathematical value, except building analysis muscles, like pumping mathematical iron.

the result of all this self flagellation, is the single theorem: the real numbers exist. wow.

god exists too, but better to just pray to her/him than worry about proving existence first.

the same holds with the reals, better to just use the axioms on faith.

still the book has a certain austere beauty, which endears it to analysis instructors.

It is written to be clear and precise, but with no consideration at all for explanation or motivation. hence it is best suited to someone who already understands it, the instructor, ass opposed to the student. It is amazing how many instructors do not realize this makes it highly unsuitable for a course text.

If you are one of the few who succeed in learning from it, you will not mind this and will always insist on inflicting it in future on your students.

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I've been learning from C. Pugh's Real Mathematical Analysis instead of from Spivak's book, but he also does Dedekind Cuts as one of the first topics, and his treatment of them is quite nice from what I remember. In my opinion, it wasn't at all laborious or unmotivated. The only sections of the book that were difficult and that left something to be desired were the later sections on Multivariate Calculus (in particular, his treatment of Stoke's Theorem is quite... terse. he really should have made it at least twice as long)

Also, I'm pretty sure there are plenty of people who will disagree with you about the existence of God. Far less than the number of people who will disagree about the mathematical validity of the real numbers.

---

bizoid: As for preparation, you might want to look at some basic stuff about metric space topology. Like prove that if you have a closed and bounded region in R^n, then it is necessarily compact and that every sequence in the region has a convergent subsequence with limit in the region. Another useful, and fairly easy to prove one is that any time you have a compact metric space, that if you have a sequence and every convergent subsequence of your sequence converges to the same value, then the entire sequence converges to that value. If you can get this far, try to read and start to understand the proof that in R^n, a sequence converges if and only if it is Cauchy. If you are able to get that far before you get to the course, then you will be good and should be able to do fine.

If you don't have that much time, then just try to learn some basic things about sequences. Limits are one of the most important tools in Analysis, and limits of sequences are one of the most common uses of a limit. Many important things in Analysis, like notions of continuity can be phrased in terms of sequences. (at least in the setting of a metric space)

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It sounds like you both recommend to have secondary course texts. (i.e. Spivak Calculus and C. Pugh)

You both have me slightly concerned now. I am a motivated student, but from your comments I feel unprepared for this course. Should I take a topology, number theory, advanced calculus or math logic course first?

Does anyone else agree with LukeD's suggestion on preparing for Analysis?

Mathwonk: Your description of the course is the exact reason why I want to take it. I just want to be cool. In addition, the Rutgers Graduate Director in Statistics recommended the course and Columbia Graduate Statistics will count this towards the credit requirements. Any thoughts on why that is?

Please keep in mind the original question:
Could you please prioritize the list from most important to review to least important to review. In addition, please provide your knowledge of the course content (e.g. professor, math grad student, student taking Analysis, student that has taken Analysis):

Review:
the course text, by Rudin
review Calc texts by Spivak / Apostall (clarification on specific topics would be helpful)
review Analysis books by Spivak / Apostall (clarification on specific topics would be helpful)
review series & sequences
review delta epsilon proofs
practice proof writing (recommendations on a text with basic proof writing problems and, more importantly, lots of detailed solutions would be appreciated.)
Thank you,
Bizoid

I'm in the minority here, but I used Rudin for a first course in analysis, and I loved it. I felt like I was LEARNING the material. You can't get through a 10-question problem set in Rudin without understanding the material.

As far as preparation goes, I think you're more than adequately prepared for the course. Unfortunately, that doesn't mean it will be easy. It will almost certainly be the most difficult thing you've ever done in your life. If you do want to brush up, brush up on limits of sequences (which are really the same thing as e-d proofs). This is the backbone of analysis, and a faultless understanding of this is absolutely essential for learning this material.

And this is for Mathwonk: we analysts feel the same way about rings of fractions. It's just torture.

mathwonk
Homework Helper
advice on dealing with fractions that i give my students: always multiply out the bottoms first.

You both have me slightly concerned now. I am a motivated student, but from your comments I feel unprepared for this course. Should I take a topology, number theory, advanced calculus or math logic course first?
I didn't mean to worry you. If you're motivated, then you should be fine in the course as long as you brush up a little on the definition of limits. What I meant is that if you study on your own before the course and manage to get through the topics I mentioned, then you'll certainly be fine and will have nothing at all to worry about. Personally though, that took me at least 2 1/2 months to get that far on my own while I was preparing for my Analysis course. However, it's certainly not necessary. I only did that because I didn't take a prereq for my course that was taught with Apostall's Calculus (the other people taking the course had already had a decent amount of exposure to limits and some exposure to the Darboux definition of the Integral, neither of which I'd done at all before preparing for the course).

Also, while I think that C. Pugh's is a good book, I've never seen any one else talk about it on this forum. Unless you get the book for your course and find that you don't think it explains things well, you shouldn't need to worry about getting another book to supplement the course text.

Since you're motivated enough to post online here and to prepare ahead of time, unless your college's course is known for being extremely difficult, you should be completely fine for the course. Most other people taking it will not be as motivated as you. Not that you shouldn't prepare, but don't worry about not being ready for it; if you work hard at it, you'll be fine.

mathwonk
Homework Helper
rings of fractions

to the analyst who disliked rings of fractions, i find it ironic that rudin assumes the existence of rational numbers , i.e. fractions, and deduces the existence of real numbers.

i.e. i made the rookie mistake of mis - stating his main theorem on reals, it is not that they exist, rather it is that reals may be constructed from the rationals, hence reals exist IF rationals exist.

but of course rationals exist only if integers exist, and integers exist only if some infinite set exists, which of course it does not, at least not in this world.

but i digress,

to understand rings one must distinguish three classes of elements: zero divisors, non zero divisors, and units. of course units are a subclass of non zero divisors.

the idea of rings of fractions, is to enlarge a given ring until certain subsets of nonzero divisors become units.

the primordial model is just the rational numbers, in which one makes all non zero divisors in the integers into units.

in general, if S is any multiplicatively closed subset of non zero divisors in a ring R, one creates a larger ring R(S) as all fractions x/y where x is in R and y is in S, with the convention that x/y = z/w iff xw = yz.

more generally, given any multiplicatively closed set S, not containing zero, mod R out by the ideal I of all elements which are zero divisors with some element of S. Now in R/I, the classes of elements from S no longer contains zero divisors, and one makes the same construction as before.

if this trivial construction is torture, fifth grade must have been hell.

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It is written to be clear and precise, but with no consideration at all for explanation or motivation. hence it is best suited to someone who already understands it, the instructor, ass opposed to the student. It is amazing how many instructors do not realize this makes it highly unsuitable for a course text.
I'm not so sure those qualities disqualify something as a good course text. Provided the instructor does a good job at motivating and explaning material in lecture, it can be a good thing. For one thing, it provides incentie to attend the lectures. But more than that, I've found that books that are heavy on motivation and explanation end up useless once you've finished taking the class, whereas more concise ones tend to have long shelf-lives as useful references. Ideally you could buy one book to learn from, and another as a reference, but this is unrealistic given the cost of textbooks these days... the real downside to concise books is that they aren't suitable for self-study.

mathwonk
Homework Helper
good points.

but if you are going to be serious, have you considered dieudonne's "foundations of modern analysis"?

its infinitely better than rudin, for serious students. i.e. at least countably better.

what about an uncountably infinite better book than rudin?

Is big Rudin worth having?

I'm more in the market for a book on algebra, myself.

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Oh, I understand it perfectly well. But the way algebra is taught at the beginning graduate level, you learn rings of fractions just for the pleasure of learning rings of fractions, with no glimpse towards localization or the reasons one might want only to bestow inverses upon some of your non-zero divisors.

At least in the case of Dedekind cuts, you have some motivation. It would be like learning Lebesgue measure and never integrating or differentiating anything.

Is big Rudin worth having?

I'm more in the market for a book on algebra, myself.
Big Rudin is much tougher than little Rudin. It's elegant, that's for certain, but the sentiment that baby Rudin is good only for those who have seen the material before would be even more applicable to big Rudin.

mathwonk
Homework Helper
on further reflection, the reason i do not use well written books like spivak for reference, is that after reading them once i understand the material and remember it, so there is no need for reference to them. rudin on the other hand is different, and lang.

when the book is not well explained, repeated reference is needed.

i like big rudin much better than baby rudin, but again, it is not so well explained.

i tried for decades to understand lebesgue integration until i listened just to one introductory lecture by a friend, an analyst teaching an undergrad class, and it all seemed clear to me afterwards.

so get it explained by someone who both understands it and is trying to make it understandable. the first couple chapters of royden are well t=regarded in this sense.

lurflurf
Homework Helper
I'm in the minority here, but I used Rudin for a first course in analysis, and I loved it. I felt like I was LEARNING the material. You can't get through a 10-question problem set in Rudin without understanding the material.

It will almost certainly be the most difficult thing you've ever done in your life.
Since you seem oblivious to some issues some of us have with the baby Rudin I will share some.

1)Difficult for the sake of difficulty
BR is like wanting to cross a parking lot so you get a sack of bricks and hike through the woods and up a hill then down the other side. Not to say there are not benifits to such an approach, but one wonders if the work is well spent and how many who could benifit from a less aduous development are stoped by this obstical.
2) Not so complete
Lots of stuff is left out and not just trivial details
3) Obscure proofs
Thm there exist a function with property P
proof
consider the function f
f has property P
QED
4) Terse to a fault.
While I can like the idea of covering in 100 pages what other cover in 200 pages, paper is not so expensive (more so with BR at 25-40 cents per page with others <10 cents/page) that it justifys making the work take 4 times longer to read.
5) Not very abstract. Could be seen as a plus given the difficulty. With this much heavy lifting we shoud at least have results true for
tensors instead of vectors (yes there are forms in chapter 10)
topological spaces instead of metric spaces
Banach spaces instead of real vector spaces
Thing go way wrong when Rudin gets to Several variables/differential forms/more general integrals and No Rudins "By reading the 200 badly written pages the reader is prepared to read very badly writen pages does not hold."

In short while one should be able to understand and enjoy some aspects of Rudin, mere reading of the text is not and efficient way to achieve this.

Baby Rudin should not be the hardest thing. If it is right away read Papa Ruding, set a marathon record, or read Bourbaki in the original french (only is you do not now read french). Any of those should harder, but also more preductive.

lurflurf
Homework Helper
But the way algebra is taught at the beginning graduate level, you learn rings of fractions just for the pleasure of learning rings of fractions, with no glimpse towards localization or the reasons one might want only to bestow inverses upon some of your non-zero divisors.
There is a problem fractions are not relly graduate level so any coverge is praobly best viewed as a review not requireing motivation. Really you have had no use for inverses?
no need to solve
AX=B
no need to embed a rig in a field
no need to show that a ring is in fact a field
ie R[sqrt(n)]~R(sqrt(n))

Rings of fractions are covered in D&F, Hungerford, and almost every other beginning graduate algebra text I've seen. Perhaps they're too trivial to be taught in a grad class. My gripe is with the presentation, not the difficulty level. The generalization of fractions to fields of fractions is trivial, and does not need motivation. The motivation behind endowing some elements of a field with inverses to the exclusion of others is not apparent at first, and to my knowledge the motivation is not apparent until one encounters localization in commutative algebra/algebraic geometry.

morphism
Homework Helper
Rings of fractions are covered in D&F, Hungerford, and almost every other beginning graduate algebra text I've seen. Perhaps they're too trivial to be taught in a grad class. My gripe is with the presentation, not the difficulty level. The generalization of fractions to fields of fractions is trivial, and does not need motivation. The motivation behind endowing some elements of a field with inverses to the exclusion of others is not apparent at first, and to my knowledge the motivation is not apparent until one encounters localization in commutative algebra/algebraic geometry.
You should see the noncommutative analogue of rings of fractions! Horrible, horrible things.

I agree with lurflurf, Rudin's proofs are terribly uninsightful. A math prof in a course I took once said that there were types of proofs-- those that showed you why it was true and those got the job done. Rudin's proofs are of the latter category. The exercises are difficult because the text fails to illustrate the general method to do the proofs for each subject.

Just seeing the important theorems and their proofs is sufficient for the advanced student. But the most important thing an introductory course must do is teach the method.

Now that's fine as long as you're using the book to supplement a lecture where the professor is filling those gaps in, the only problem being that many professors teach by simply writing the textbook on the board. It's like thank you, I already own a copy of the book.