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Real analysis

What a time to be alive...before sex, drug, rock n roll, smartphones, social media, etc reduced youth's average attention span to the level worse than a gold fish's.
I second that. I have cousins that are younger who can’t look up from a screen. They have no thought process, walking around like a bunch of mindless zombies.
 

Daniel Duffy

C++ author, trainer
Anything except Rudin.
I don't disagree, but it depends on the target audience.
How would you compare and contrast Rudin versus the others?
One thing with Rudin's exercises are like "give an example of X with property Y", or "generalize zz' etc. Less spoonfeeding than the others?
Example: exercise 6 on page 39 .. it is meant in a sense to train future PhD students in mathematical thinking.

In the preface it has a disclaimer "... by advanced undergraduate or by first-year graduate students who study mathematics". Loud and clear.
Most people have glossed over the preface and probably misunderstood the requirements.
 
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How would you compare and contrast Rudin versus the others?

(Baby) Rudin has a cramped and terse style which inhibits learning. Ross, Abbott, and Bartle & Sherbert are all good books to learn from. For things like the implicit function theorem and the inverse function theorem (not covered in these books) I would use an advanced calculus book from the '60s like Taylor. For the Lebesgue integral, something else again. Rudin is 65 years old and besides its age has nothing to recommend it. Don't get me wrong: I like old books, which often have a style and a coverage the more modern texts lack. But I can't say this of Rudin. Even Rudin's Real and Complex Analysis would be better replaced by the four books by Stein and Shakarchi.
 
Why then, does Rudin get recommended?

*Shrug* -- inertia and "CYA" (which means "cover your ass"). No academic is going to get into trouble for assigning Rudin as the text. Also keep in mind that the average age of professors in the USA keeps increasing -- you're likely to be taught by some professor in his 60s, who used the text forty years ago as an undergrad. Even if the prof isn't teaching the course, he may be the one who chooses the book.
 

Daniel Duffy

C++ author, trainer
LOL; those guys are younger than I am. It is never too late to change your ways.
Seriously, tho' ... we used Rudin as background and we relied on the excellent teaching skills of our prof to show how to solve problems. And then taking initiative (by reading other books) to understand the topics.

I think trying to learn Real Analysis by studying Rudin on your own would be a depressing experience.

A more up to date remark is that (applied) functional analysis and numerical analysis is a better investment.

Anyways, that's my 2 cents.

//

Paul Halmos was a great teacher

...the source of all great mathematics is the special case, the concrete example. It is frequent in mathematics that every instance of a concept of seemingly great generality is in essence the same as a small and concrete special case.
I want to be a Mathematician, (Washington 1985).
 

Daniel Duffy

C++ author, trainer
Sanity 101 check, test for Lebesgue and measurable sets understanding..

Integrate f(x)=x(1−x) on (0,1) by hand from 1st principles (upper and lower sums based on disjoint(?) measurable sets) with say 4 divisions of
0≤f(x)≤1.
Using pen and paper.

It even can be automated but you need to find all roots xx of Y=f(x) where Y is a given function value. In this case it is a quadratic equation.

Put icing on the cake by programming it in C++. Once that's working then you can generalize it to any f(x). Then f(x,y)..

This teaches both real and numerical analysis. Aka Halmos' concrete example.
 
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LOL; those guys are younger than I am. It is never too late to change your ways.

You live in Europe. I don't think you know how ossified US universities have become over the last few decades. In subjects like real analysis, complex analysis, linear algebra, and perhaps abstract algebra often the same texts and the same lectures are being employed as they were forty years ago. Again, don't get me wrong: I'm enamored of some old texts. In complex analysis, for example, I particularly like the texts by Whittaker and by Copson as they explain what the complex integral is in terms of Riemann sums (something most modern books adroitly elide over). But departments are still using Ahlfors after half a century?
 

Daniel Duffy

C++ author, trainer
Back then, the approach was 14 lectures on complex analysis in 2nd year and in 3rd year conformal mapping for fluid dynamics, several complex variables.

Flashback: saw cover of Copson on Amazon .. it's the great book we used.
 
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Daniel Duffy

C++ author, trainer
You live in Europe. I don't think you know how ossified US universities have become over the last few decades. In subjects like real analysis, complex analysis, linear algebra, and perhaps abstract algebra often the same texts and the same lectures are being employed as they were forty years ago. Again, don't get me wrong: I'm enamored of some old texts. In complex analysis, for example, I particularly like the texts by Whittaker and by Copson as they explain what the complex integral is in terms of Riemann sums (something most modern books adroitly elide over). But departments are still using Ahlfors after half a century?
Riemann sums .. nice that it is essentially the Euler method. And it is computable.
 

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Here are a few book suggestions for analysis.

1 Understanding Analysis - Abbott
2 How to think about analysis - Alcock
The above two books are simply awesome to get an intuitive grasp of elementary real analysis. These books are also perfect before tackling the rigorous Rudin's mathematical analysis.

Complex Analysis
Visual complex analysis - Needham is hard to beat.

Functional Analysis
Introduction to functional analysis & applications - Kreyszig.
Topology & modern analysis - Simmons.
 
If you just wanna study the stuff(no certification), there are some lecture videos posted on youtube by Haverford and other colleges. In terms of the preq, you should be familiar with the mathematical notations and some basic proof logics. Basically, the video will cover materials up to chap 6 in Rudin, which will be basic topology, sequence and series convergence, continuity, differentiation and Riemman-Stieltjes integral. Yet, I haven't seen any video covered uniform convergence, Weinstrass theorem and Lebesgue theory, which is from chap 7 - 11 in Rudin. So probably, you have to study those stuffs by yourself(those are the core materials of baby analysis).


Francis Su of Harvey Mudd also has a one semester course using Rudin online.
 
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