Does my Bachelor's have enough mathematics?

Daniel Duffy

C++ author, trainer
What % of incoming MFE students do you think even take real analysis? I’d venture to say less than 10%. If you’re going to learn real analysis, Rudin is certainly not a bad candidate to start with. Like I said, Pugh and Zorich also have written good analysis books, but in my opinion Rudin is a must read book for math majors and is still very much present in honors UG math real analysis sequences in the US.

Real and complex analysis by Rudin is not UG level, it largely functions as a first year PhD text for a real/complex analysis sequence.

And Schaum’s outlines real analysis instead of Rudin? Surely this must be a joke?
We can agree to differ. I'm not joking about Schaum. Have you seen that book or is it a knee-jerk reaction?
I did Rudin R & C in 2nd year undergrad :) (in Europe)
 
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Daniel Duffy

C++ author, trainer
Papa Rudin 2nd year of UG? I should probably stop arguing with you then! That is incredible.
It was a special honours degree in general. Actually, the dept prof gave those courses .. he was a PhD student of William Feller at Princeton so we were set high targets.
3rd and 4th years were even more intensive :) 50 students entered in year 1, 6 finished in year 4.
 
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Daniel Duffy

C++ author, trainer
This book contains the standard 1st year graduate material on Real Analysis.A very solid background in undergraduate analysis is required to get started in reading and working through this book.In the United States one rarely finds a graduate level math textbook with solved problems unfortunately.This interesting book is one of the few exceptions to this rule.The 375 solved problems in this outline are most instructive and an essential tool for anybody that truly wants to understand what is really going on in this difficult subject.There is also plenty of supplementary problems for the reader to try on his/her own.One of the most interesting aspects of this book is the development of the Lebesgue Integral in terms of Upper and Lower sums .Too often people do not understand the more abstract definition given in just about all other textbooks (*).This definition is parallel with that of the Riemann Integral from undergratuate anaylsis.This book is a true gem and a rare find.

https://images-na.ssl-images-amazon.com/images/I/51kyIlmZr5L._SX354_BO1,204,203,200_.jpg

(*) numerical analysts like this approach
 
It's good that you are keeping backups. MHTCET provides a lot of decent colleges trying to get into a maths heavy course. Something like Mathematics and computing is tailor-made for QF. Also, choose a college where banks and quant firms come. There are many colleges apart from IITs where GS comes for SDE roles also some quant firms like Graviton and Alphagrep visit other colleges. If ur in Mumbai JPMC can give u a good opportunity as well. They hire a lot from outside IITs for risk management and SDE roles. Idk where MS visits.
Thanks for the advice I'll sure do look out for it.
Good luck to u as well.
Yeah i am actually targetting VJTI and SPIT colleges where GS and JPMC come to recruit... GS gives only software role whereas JPMC gives Quant role in SPIT
 
I recently found a great resource from a quant researcher at Citadel Securities which re-emphasizes some of my previous points...

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Daniel Duffy

C++ author, trainer
Given it is an introductory book for real analysis, all of the topics covered are pre 1920s/1930s and really pre 1900s given there is only one chapter on Lebesgue integration. It is very much the same today, it is a great first analysis book for those students looking to do a PhD in math in my opinion.
Unfortunately, that is not so. Been there, done that.

OK, I had a look again at baby Rudin, in particular chapter 10. Long story short ... it does a great disservice/injustice to the topics of Measure Theory and Lebesgue integration. We used our lecture notes as basis for the real thing.

Speaking didactedl and from a learning perpective, the main issues are IMO:

1. Lack of concrete/numerical examples throughout. This is very bad.

2. 3 pages devoted to Measure Theory.

3. No distinction between bounded and unbounded function.

4. Only 2 theorems discussed: no examples.

5. No 2d Lebesgue theory.

6. Missed opportunity to couple with chapter 3 (integration of series and sequences).

7. A better comparison would have been: Lebesgue versus Riemann-Stieltjes (upper and lower limits).

(8. I find it a boring read).

Spiegel's Schaum book and my lectures at the the time fill in these gaps.

Learning from Rudin is a bridge too far for many.


(BTW Paul Halmos wrote a great book on Measure Theory)
quote-the-source-of-all-great-mathematics-is-the-special-case-the-concrete-example-it-is-frequ...jpg
 
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Daniel Duffy

C++ author, trainer
This book contains the standard 1st year graduate material on Real Analysis.A very solid background in undergraduate analysis is required to get started in reading and working through this book.In the United States one rarely finds a graduate level math textbook with solved problems unfortunately.This interesting book is one of the few exceptions to this rule.The 375 solved problems in this outline are most instructive and an essential tool for anybody that truly wants to understand what is really going on in this difficult subject.There is also plenty of supplementary problems for the reader to try on his/her own.One of the most interesting aspects of this book is the development of the Lebesgue Integral in terms of Upper and Lower sums .Too often people do not understand the more abstract definition given in just about all other textbooks (*).This definition is parallel with that of the Riemann Integral from undergratuate anaylsis.This book is a true gem and a rare find.

https://images-na.ssl-images-amazon.com/images/I/51kyIlmZr5L._SX354_BO1,204,203,200_.jpg

(*) numerical analysts like this approach
A good exercise is to compute the Lebesgue integral NUMERICALLY of y = f(x) by hand from 1st principles, i.e. upper and lower sums.
By doing this, you also learn what a level set is and how Lebesgue differs from Riemann.

e.g. y = x on (0,1), exact = 1/2.
 
And for incoming MFE students Rudin #1 is not suitable. Cruel.

That Rudin is still being used just goes to show how ossified US math departments are. My theory is you have all these 60- and 70-something math professors in US universities using the book out of sheer inertia, and because it's what they used in their youth. The same for using Ahlfors for complex analysis. There are so many better alternatives to real analysis (and complex analysis) today -- and have been for decades. Off the top of my head, Garling's three volumes on analysis, published by Cambridge, and used at Cambridge by second- and third-year undergrads, is an immeasurably superior alternative.
 

Daniel Duffy

C++ author, trainer
I'll be 70 in a few weeks and even I knew in 1972 that Rudin was not useful.
I wonder how many students get burned on Dedekind cuts (chapter 1)......

If Rudin is all you know then there is little hope to understand more advanced and applied topics.
 
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Baby Rudin is not a measure theory book (I learned my measure theory from Folland, Royden, and Stein/Shakarchi, all respectable sources), and again, is used as a first analysis text not the only analysis text. I really do not think it is a bad place to start. If it were, why would top institutions in the US continue to use it in their honors sequences? I don’t think it is simply because they used it for their training, I think it is because it simply is the most rigorous intro analysis book on the market. Again, for the third time, I also used Zorich and Pugh’s texts which are phenomenal resources. Like you say, horses for courses. Baby Rudin is not a bad resource for one’s first encounter with analysis — I firmly stand by this point.

Ahlfors is admittedly dated. I personally like Stein/Shakarchi, but Ahlfors is a standard (to my knowledge) reference for most qualifying exams covering complex analysis.

Point: Rudin and Ahlfors are not nearly as bad as you both are making them out to seem.
 
A good exercise is to compute the Lebesgue integral NUMERICALLY of y = f(x) by hand from 1st principles, i.e. upper and lower sums.
By doing this, you also learn what a level set is and how Lebesgue differs from Riemann.

e.g. y = x on (0,1), exact = 1/2.
This is a standard homework exercise regardless of the text used…
 
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Other than Garling's three volumes, there's also the thre volumes by Amman and Escher, which can again be unequivocally recommended. And also the books by David Bressoud. Zorich is fine as well. I just don't see the rationale in using a book which came out when Elvis Pressley was singing "You ain't nothin' but a hound dog."

Postscript: And the three or four volumes by Stein and Shakarchi are also fine.
 

Daniel Duffy

C++ author, trainer
Baby Rudin is not a measure theory book (I learned my measure theory from Folland, Royden, and Stein/Shakarchi, all respectable sources), and again, is used as a first analysis text not the only analysis text. I really do not think it is a bad place to start. If it were, why would top institutions in the US continue to use it in their honors sequences? I don’t think it is simply because they used it for their training, I think it is because it simply is the most rigorous intro analysis book on the market. Again, for the third time, I also used Zorich and Pugh’s texts which are phenomenal resources. Like you say, horses for courses. Baby Rudin is not a bad resource for one’s first encounter with analysis — I firmly stand by this point.

Ahlfors is admittedly dated. I personally like Stein/Shakarchi, but Ahlfors is a standard (to my knowledge) reference for most qualifying exams covering complex analysis.

Point: Rudin and Ahlfors are not nearly as bad as you both are making them out to seem.
For the record, I said nothing about Ahlfors, just Rudin.
 

Daniel Duffy

C++ author, trainer
not to say there aren’t pieces that are constructive
that's Double Dutch :whistle:

I can't proceed with a reply.

the nature of the subject
is it?
 
not to say there aren’t pieces that are constructive
that's Double Dutch :whistle:

I can't proceed with a reply.

the nature of the subject
is it?
Real analysis is not entirely non-constructive… better? Double/triple negatives are not uncommon in proof writing…

Yes. Measure theory in particular is by and large very non-constructive. You in fact were who made an even broader claim: “Most real analysis is non-constructive...”
 
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