How to study mathematical analysis?

[Arian.D]

I'm not quite sure if my question belongs to this section or not, so I hope I would be posting it in the right place. Anyway, I have troubles understanding Rudin's principals of mathematical analysis. To me, everything he says, is absurd. I can't understand what he says, neither do I want to understand what he says I may say.

You know, when I study mathematics, I like to have an intuition of what we're talking about before we go into technical details and abstraction. To me, a neat mathematical idea, is formed when you have gained a correct intuition and insight into the material and only after that you could care about rigor. In Rudin, he roughly explains anything. It's all theorems and definitions. It's just like he has gathered a bunch of proofs of some easy theorems proved in the hardest way and have called that a book. For example in Chapter 6, when he talks about Riemann-Stieljes integrals he assumes α (the integrator) is an increasing function. While in Apostol's analysis, Apostol doesn't require the integrator function to be increasing. So in my opinion, even though I don't know much about the subject, Rudin is just putting an unnecessary extra condition on the integrator and not only it doesn't make proofs easier, but it makes them quite tedious and hard.

To cut the ranting short, I'm looking for a book that discusses mathematical analysis in a nice and understandable way. Not like Rudin that explains the simplest stuff in the ridiculously hardest way. I should add that my main area of interest, as for now, is abstract algebra. But my ultimate goal is to be a geometer that understands concepts like manifolds and other structures that are related to our understanding of the universe around us. So I'm sure I'll need to know analysis fairly well if I want to understand smooth mappings and other analytical concepts later in differential geometry and elsewhere. Having said that, please take into account that I don't have so much time to spend on an analysis book now because my university courses are really exhausting (I'm taking Calculus III, an introduction to module-theory, numerical analysis I, analysis II, topology and ordinary differential equations this semester, so I'm pretty busy with other subjects and I can't devote all my time to analysis :( ).

Any helps would be appreciated. And I'm sorry If I've posted my question in the wrong place.

 

[HallsofIvy]

It seems odd to me that you say "I don't know much about the subject" yet you assert that Rudin is wrong. In my opinion Rudin one of the clearest, simplest (perhaps a little oversimplified) accounts of analysis going. Certainly far simpler than Apostol! Rudin includes the requirement that measure function be increasing because without that it wouldn't be a measure and you wouldn't have an integral! Apostol does, in fact, make that requirement, perhaps you just missed it.

 

[jgens]

Anyway, I have troubles understanding Rudin's principals of mathematical analysis. To me, everything he says, is absurd. I can't understand what he says, neither do I want to understand what he says I may say.

I think Rudin is actually fairly clear, so if you are having trouble understanding I would say that this is likely due either to insufficient preparation for the text (which probably is not the case if you are studying abstract algebra now) or is a symptom of not spending enough time working through the text and really thinking about the proofs. Being a university student myself, I understand not having sufficient time to work through everything sufficiently; in these cases, I have found that this is what the summer is best used for.

You know, when I study mathematics, I like to have an intuition of what we're talking about before we go into technical details and abstraction. To me, a neat mathematical idea, is formed when you have gained a correct intuition and insight into the material and only after that you could care about rigor.

Everyone learns a little differently and you will rarely have a textbook suited exactly to your needs. There is value in having someone explain the intuition behind the concepts and then working out the theory, but there is also value in working out the theory and having you work out the intuition. In the future, you will inevitably have to work through another book like Rudin, so it is worthwhile getting used to it now when the material is easy.

In Rudin, he roughly explains anything. It's all theorems and definitions. It's just like he has gathered a bunch of proofs of some easy theorems proved in the hardest way and have called that a book.

To be fair, most students working through Rudin have seen the material before in some way or another; in fact, any decent calculus course should essentially cover the material in the first 7 chapters of his book. It is also worth noting that Rudin is (in my opinion) a fairly gentle introduction to analysis. You could be working through a text like Kolmogorov and Fomin instead.

If you happen to be talking about the later chapters of Rudin (particularly chapters 10 and 11), I will agree with you that they can be a little confusing on a first read through. Part of this has to do with the difficulty of defining differential forms in an elementary manner and without the ability to appeal to concepts from algebra. The tools that make differential forms much more tractable tend to come from abstract algebra.

For example in Chapter 6, when he talks about Riemann-Stieljes integrals he assumes α (the integrator) is an increasing function. While in Apostol's analysis, Apostol doesn't require the integrator function to be increasing. So in my opinion, even though I don't know much about the subject, Rudin is just putting an unnecessary extra condition on the integrator and not only it doesn't make proofs easier, but it makes them quite tedious and hard.

The requirement that your integrator be increasing should actually simplify the proofs since your integrator will be better behaved. In any case, one way to motivate choosing α an increasing function stems from measure theory and Lebesgue integration. Another motivation stems from making the existence theorems easier; that is, proving that a Riemann-Stieljes integral exists under the appropriate hypotheses.

To cut the ranting short, I'm looking for a book that discusses mathematical analysis in a nice and understandable way.

I have heard good things about Pugh if you are interested in looking into his book.

 

[zooxanthellae]

You could be working through a text like Kolmogorov and Fomin instead.

For what it's worth, I'm currently finishing up Real Analysis and I preferred Kolmogorov and Fomin to Rudin. It's similarly rigorous but for whatever reason I find it much clearer than Rudin.

I have heard good things about Pugh if you are interested in looking into his book.

Pugh is certainly a believer in intuition and it shows through in his book. He uses a lot of pictures and intuitive arguments to do things and some people really like this approach. I found it at times unpleasantly vague and confusing and preferred K&F, but it sounds like you would like it more.

 

[SteveL27]

Any helps would be appreciated. And I'm sorry If I've posted my question in the wrong place.

FWIW perhaps you have a bad teacher. Reason I say that is that I happened to have had a truly marvelous teacher for that class. Used baby Rudin as well. But for that class a good teacher/guide is essential. Because you're right, it's a lot of detail about things we normally take totally for granted.

What is the course about? It's about clarifying, once and for all, these vague concepts of "infinitely close" that permeate calculus.

When we defined the derivative as the limit of the difference quotient, what are we saying? For any nonzero difference between x and x^' (or x and x+h, if you prefer that formulation), the difference quotient is just the quotient of two nonzero real numbers.

Now we say, let x^' "approach" x (or let h approach zero). Then voila, here's this thing called a limit.

But there's a humongous logical problem here. If h is not zero, then x and x^' are different numbers, so the quotient is defined, but it's still not the "limit," whatever that is.

But if x = x^' (or h = 0) then we have 0/0. And that's undefined.

So what is this "limit" thing?

This is a deep and profound question. Newton didn't know the answer, though his writings show that he certainly understood the problem.

Bishop Berkeley (pronounced "Barkley" like the former basketball player) was a critic of calculus and wrote a pamphlet pointing out these logical flaws. He called the derivative (at that time called a "fluxion") the "ghost of departed quantities."

Newton worked and published in the 1680's or so. And the notion of limit was not finally, correctly, logically explained and defined, until the 1850's. That's 170 years, more or less, to answer the question of what the difference quotient actually is.

When you study real analysis, you are being taught the end result of 170 years of work by the greatest mathematicians in the world, struggling to explain how the difference quotient "becomes" the limit.

But the textbooks never EXPLAIN it this way. They never give people a view of the intellectual struggle to explain what a limit is. All they do is dive into the details: "A Dedekind cut is such and so." Then for the rest of the book they never talk about Dedekind cuts. So why'd they bother?

The answer is that when we talk about the real numbers in terms of their properties, we need to know that such a collection of mathematical fictions can be constructed out of the rules of logic, along with whatever objects we happen to already believe in: in this case, the rationals.

So first they prove that there IS such a thing as the real numbers.

Then they define sequences of real numbers ... and Cauchy sequences ... and voila, suddenly they have a mathematical definition of what it means for a sequence to have a limit. A definition that the purest, most critical logician in the world would have to agree is air tight.

I hope this wasn't too long and rambling, but that's what I think this course is about. And I wish that in real analysis class and also in every other class, they did a better job of placing the math in a historical context; so that students can appreciate that it took people a long time to understand what we take for granted today.

And if your teacher's not helping, find a TA and get them to work with you. It helps to have a friendly and capable guide through this material.

Sure hope this helps.

I'm taking Calculus III, an introduction to module-theory, numerical analysis I, analysis II, topology and ordinary differential equations this semester, so I'm pretty busy with other subjects and I can't devote all my time to analysis :( ).

ps -- I hope my thoughts are helpful to someone who might be taking the course, but after re-reading your post I came across your class load.

You are taking way way too much this semester. If you did NOTHING but real analysis, you could spend all your time on it. It's a critical course in one's mathematical progress. Maybe THE critical course. Not just the material, but the way of thinking.

It's simply not possible to take real analysis with all these other heavy courses. So forget what I said about the history and drop half of those courses. That would be my advice.

 

[Arian.D]

It seems odd to me that you say "I don't know much about the subject" yet you assert that Rudin is wrong. In my opinion Rudin one of the clearest, simplest (perhaps a little oversimplified) accounts of analysis going. Certainly far simpler than Apostol! Rudin includes the requirement that measure function be increasing because without that it wouldn't be a measure and you wouldn't have an integral! Apostol does, in fact, make that requirement, perhaps you just missed it.

I didn't assert Rudin was wrong. I said he likes to make simple things hard. Excluding some theorems that can't be proved without having a knowledge of topology like the intermediate value theorem and the theorem which proves the existence of upper and lower bounds for any continuous function on a closed interval I've proved other concepts and theorems in Calculus I before. So it's not like I have no idea about analysis at all. About Apostol, I found it much clearer than Rudin. He defines things very easily and when you read his book you learn new things. He also gives a lot of examples to make the concepts more obvious. If the professor in my class didn't use Rudin I would've certainly used Apostol instead.

About the assumption that alpha [the integrator] must be an increasing function, I'm sure you're right and it might later become clear for me why Rudin is assuming alpha to be increasing when I study measures, but about Apostol, I checked the book and I'm quoting what Apostol claims here:

For brevity we make certain stipulations concerning notation and terminology to be used in this chapter. We shall be working with a compact interval [a,b,] and, unless otherwise stated, all functions denoted by f,g,α,β, etc., will be assumed to be real-valued functions defined and bounded on [a,b]

So he says alpha needs to be a bounded real-valued function, he never mentions it needs to be increasing.

He defines Riemann-Stieltjes integral in the following way:

Definition 7.1. Let $P=\{x_0,x_1,...,x_n\}$ be a partition of $[a,b]$ and let $t_k$ be a point in the subinterval $[x_{k-1}, x_k]$. A sum of the form

$$S(P,f,\alpha) = \sum_{k=1}^n{f(t_k)\Delta \alpha_k}$$​

is called a Riemann-Stieltjes sum of $f$ with respect to $\alpha$. We say f is Riemann integrable with respect to $\alpha$ on $[a,b]$, and we write "$f\in R(\alpha)$ on $[a,b]$" if there exists a number $A$ having the following property:

For every $\epsilon>0$, there exists a partition $P_{\epsilon}$ of $[a,b]$ such that for every partition P finer than $P_{\epsilon}$ and for every choice of the points $t_k$ in $[x_{n-1}, x_n]$, we have
$$|S(P,f,\alpha) - A| < \epsilon$$

This definition has several advantages. First of all it's very easy to verify that if the integral exists it's unique. With Rudin, it's not very easy to prove that the integral is unique unless we have already proved that sup and inf of a subset of Real numbers is defined uniquely (correct me if I'm wrong).

Then it becomes very easy to show that a Riemann-Stieltjes integral is linear in respect to both the integrand and the integrator. Simple arithmetic with inequalities and using triangle inequality results that. While if you want to prove this with Rudin's definition, first you'll need to show that the sums are each integrable on [a,b] and then you'll need to prove that if you multiply the integrand by a constant the result is still integrable and stuff like that which means you'll have to show that the upper integral and the lower integral are equal which is quite tedious. Apostol uses a very neat form of Riemann's criterion for integrability from the very beginning which makes it very easy to prove some obvious properties of integration. Rudin too proves Apostol's definition (with the assumption that alpha is increasing) as a theorem but he rarely uses it later in the chapter if I'm not mistaken.
I prefer Apostol to Rudin, and when I read this book I feel much more comfortable. When I read Rudin I feel like I'm just wasting my time with some mathematical definitions :(.

 

[Arian.D]

FWIW perhaps you have a bad teacher. Reason I say that is that I happened to have had a truly marvelous teacher for that class. Used baby Rudin as well. But for that class a good teacher/guide is essential. Because you're right, it's a lot of detail about things we normally take totally for granted.

What is the course about? It's about clarifying, once and for all, these vague concepts of "infinitely close" that permeate calculus.

When we defined the derivative as the limit of the difference quotient, what are we saying? For any nonzero difference between x and x^' (or x and x+h, if you prefer that formulation), the difference quotient is just the quotient of two nonzero real numbers.

Now we say, let x^' "approach" x (or let h approach zero). Then voila, here's this thing called a limit.

But there's a humongous logical problem here. If h is not zero, then x and x^' are different numbers, so the quotient is defined, but it's still not the "limit," whatever that is.

But if x = x^' (or h = 0) then we have 0/0. And that's undefined.

So what is this "limit" thing?

This is a deep and profound question. Newton didn't know the answer, though his writings show that he certainly understood the problem.

Bishop Berkeley (pronounced "Barkley" like the former basketball player) was a critic of calculus and wrote a pamphlet pointing out these logical flaws. He called the derivative (at that time called a "fluxion") the "ghost of departed quantities."

Newton worked and published in the 1680's or so. And the notion of limit was not finally, correctly, logically explained and defined, until the 1850's. That's 170 years, more or less, to answer the question of what the difference quotient actually is.

When you study real analysis, you are being taught the end result of 170 years of work by the greatest mathematicians in the world, struggling to explain how the difference quotient "becomes" the limit.

But the textbooks never EXPLAIN it this way. They never give people a view of the intellectual struggle to explain what a limit is. All they do is dive into the details: "A Dedekind cut is such and so." Then for the rest of the book they never talk about Dedekind cuts. So why'd they bother?

The answer is that when we talk about the real numbers in terms of their properties, we need to know that such a collection of mathematical fictions can be constructed out of the rules of logic, along with whatever objects we happen to already believe in: in this case, the rationals.

So first they prove that there IS such a thing as the real numbers.

Then they define sequences of real numbers ... and Cauchy sequences ... and voila, suddenly they have a mathematical definition of what it means for a sequence to have a limit. A definition that the purest, most critical logician in the world would have to agree is air tight.

I hope this wasn't too long and rambling, but that's what I think this course is about. And I wish that in real analysis class and also in every other class, they did a better job of placing the math in a historical context; so that students can appreciate that it took people a long time to understand what we take for granted today.

And if your teacher's not helping, find a TA and get them to work with you. It helps to have a friendly and capable guide through this material.

Sure hope this helps.

Well, I've read first chapters of Newton's classical text "Methods of Fluxions". He talks about Fluxions and Fluents but his main area of interest lies in infinite series and geometric properties of curve. What he talks about in that book is very different than what we today study in Calculus.
Euler's "foundations of differential calculus" is much closer to our today's way of dealing with Calculus than Newton's "methods of fluxions". I've read the first several chapters of that book too, and other chapters are still to be published by springer if I'm not mistaken.
When Euler died, he had left hundreds of unanswerable questions for mathematicians. As it is obviously seen in Euler's "Elements of Algebra" and "Analysin infinitorum" Euler didn't care about the convergence of series and he thought that all principles of Algebra that are correct when we deal with finite situations could be generalized to infinite situations. Cauchy was the first person to criticize Euler's works in a rigorous way, he rejected generality of algebra that was assumed by mathematicians like Euler and Lagrange and he started to add rigor to their works. You know, my problem is that I prefer Euler's way. Euler showed us the beauty of mathematics and his deep understanding of mathematical beauty is stated in the simplest way possible in the formula $e^{\pi i}+1=0$, he advanced mathematics in a way that possibly no one else did and even when he committed mistakes, his mistakes opened new ideas in mathematics like divergent series. That's why I can't get along with Rudin's analysis. He kills your mathematical creativity. When you read Rudin's mathematical analysis, you learn the theory, but you don't learn to apply the theory you have learned in real situation problems. That's my problem with Rudin and books of that sort.

ps -- I hope my thoughts are helpful to someone who might be taking the course, but after re-reading your post I came across your class load.

You are taking way way too much this semester. If you did NOTHING but real analysis, you could spend all your time on it. It's a critical course in one's mathematical progress. Maybe THE critical course. Not just the material, but the way of thinking.

It's simply not possible to take real analysis with all these other heavy courses. So forget what I said about the history and drop half of those courses. That would be my advice.

Yea. Next semester I will try to take less courses to have more time to study them in details. :(

 

[Arian.D]

I think Rudin is actually fairly clear, so if you are having trouble understanding I would say that this is likely due either to insufficient preparation for the text (which probably is not the case if you are studying abstract algebra now) or is a symptom of not spending enough time working through the text and really thinking about the proofs. Being a university student myself, I understand not having sufficient time to work through everything sufficiently; in these cases, I have found that this is what the summer is best used for.
Everyone learns a little differently and you will rarely have a textbook suited exactly to your needs. There is value in having someone explain the intuition behind the concepts and then working out the theory, but there is also value in working out the theory and having you work out the intuition. In the future, you will inevitably have to work through another book like Rudin, so it is worthwhile getting used to it now when the material is easy.
To be fair, most students working through Rudin have seen the material before in some way or another; in fact, any decent calculus course should essentially cover the material in the first 7 chapters of his book. It is also worth noting that Rudin is (in my opinion) a fairly gentle introduction to analysis. You could be working through a text like Kolmogorov and Fomin instead.

If you happen to be talking about the later chapters of Rudin (particularly chapters 10 and 11), I will agree with you that they can be a little confusing on a first read through. Part of this has to do with the difficulty of defining differential forms in an elementary manner and without the ability to appeal to concepts from algebra. The tools that make differential forms much more tractable tend to come from abstract algebra.
The requirement that your integrator be increasing should actually simplify the proofs since your integrator will be better behaved. In any case, one way to motivate choosing α an increasing function stems from measure theory and Lebesgue integration. Another motivation stems from making the existence theorems easier; that is, proving that a Riemann-Stieljes integral exists under the appropriate hypotheses.
I have heard good things about Pugh if you are interested in looking into his book.

Differential forms are hard, but then you need to have a fairly well understanding of linear algebra and ideas in differential geometry to understand them completely I think. I'm not complaining because I don't understand Rudin's mathematical analysis, I'm complaining because Rudin doesn't teach you anything. He just proves and proves, no intuitive explanations, no examples, nothing! And when you try to solve the exercises in each chapter, you find them devilishly hard which is very disappointing for me.
If it weren't for the final exam, I would've thrown this book in the dust been sooner than I open this thread. But the problem is that our professor will use Rudin's proofs and exercises in the final exam, so I can't do that now. I don't want to memorize proofs, I want to learn the strategy. I'm sure it's my own fault too because I'm not spending enough time on analysis this semester, but still, I'm having troubles understanding Rudin's proofs without having intuition and I don't want to memorize the proofs either, because that goes against my philosophy of learning mathematics. So I'm really concerned that I might get a bad grade in the final exam :(

If you write the complete names of the books you mentioned, I'll be very appreciative.

For what it's worth, I'm currently finishing up Real Analysis and I preferred Kolmogorov and Fomin to Rudin. It's similarly rigorous but for whatever reason I find it much clearer than Rudin.
Pugh is certainly a believer in intuition and it shows through in his book. He uses a lot of pictures and intuitive arguments to do things and some people really like this approach. I found it at times unpleasantly vague and confusing and preferred K&F, but it sounds like you would like it more.

Well, I guess I'd like Pugh, but again I need the complete name to look for it in my university's library.

P.S: I'm sorry if I'm posting several replies here, I thought if I wanted to answer all those replies in a single post my post would get long and terribly unclear for the readers.

 

[micromass]

I don't really like Rudin either. I think Rudin is a great and wonderful book, but only for those people who already know analysis. If you're not comfortable with it yet, then you won't like it.

Here are some books you should check out:

Abbott:
https://www.amazon.com/dp/0387950605/?tag=pfamazon01-20

Apostol:
https://www.amazon.com/dp/0201002884/?tag=pfamazon01-20

Bartle:
https://www.amazon.com/dp/0471433314/?tag=pfamazon01-20
https://www.amazon.com/dp/047105464X/?tag=pfamazon01-20

Carothers:
https://www.amazon.com/dp/0521497566/?tag=pfamazon01-20

Pugh:
https://www.amazon.com/dp/144192941X/?tag=pfamazon01-20

I like all these books much more than Rudin. I'm sure you'll find some books you'll like.

 

[Arian.D]

I'm starting to fall in love with Charles Chapman Pugh's analysis book. It's exactly what I want to learn analysis from, but there's a problem, a big problem I must say. It doesn't cover the topics that we'll cover in my university's Analysis II course. We'll cover Riemann-Stieltjes integration, functions of bounded variations and sequences of functions, and special functions (like sinx,cosx, gamma function, etc).
Fortunately Rudin explains transcendental functions in a nice way. I like how Rudin treats gamma function and other non-algebraic functions like sine and cosine in chapter 8. Although I personally prefer to define sinx and cosx as functions satisfying two ODE's that are related to each other, but Rudin's way of treating these functions isn't as ugly as how it treats Riemann-Stieltjes integration for example.

So, I could say that I loved Pugh's book, but it won't help me to get a good grade in the final exam. Having the style of Pugh's book in mind, what other books could I benefit from to study the topics I said? (Riemann-Stieltjes integration, functions of bounded variation, sequences of functions and non-algebraic functions).

 

[micromass]

I'm starting to fall in love with Charles Chapman Pugh's analysis book. It's exactly what I want to learn analysis from, but there's a problem, a big problem I must say. It doesn't cover the topics that we'll cover in my university's Analysis II course. We'll cover Riemann-Stieltjes integration, functions of bounded variations and sequences of functions, and special functions (like sinx,cosx, gamma function, etc).
Fortunately Rudin explains transcendental functions in a nice way. I like how Rudin treats gamma function and other non-algebraic functions like sine and cosine in chapter 8. Although I personally prefer to define sinx and cosx as functions satisfying two ODE's that are related to each other, but Rudin's way of treating these functions isn't as ugly as how it treats Riemann-Stieltjes integration for example.

So, I could say that I loved Pugh's book, but it won't help me to get a good grade in the final exam. Having the style of Pugh's book in mind, what other books could I benefit from to study the topics I said? (Riemann-Stieltjes integration, functions of bounded variation, sequences of functions and non-algebraic functions).

Did you check Carothers his book?? It contains a discussion on those topics. Also it discusses Riemann-Stieltjes integrals where the integrator is not necessarily increasing.

 

[iceblits]

I think I'm in the same sort of boat as you. I'm using Rudin as we speak and just finished chapter one. I think the trick is to not simply read the proofs following the theorem...rather...try to prove it before reading...its actually very rewarding. I'm also using the Harvey Mudd college real analysis playlist on youtube (incidently, they use Rudin) so it ties together nicely.