ODE and Analysis book recommendations

Quasar Chunawala

Active Member
Hi friends,

(1) I am using the book on Linear Analysis by Kreider, Kuller, Ostberg and Perkins in my differential equations course, alongside Ordinary Differential Equations by Tenenbaum and Pollard. I really like the author's use of linear operators to explain the intuition behind the results in the theory of differential equations. Are there any other good books that you would recommend, are of practical value while studying ODEs/PDEs?

It'd be nice, if I can get a couple of good suggestions to master the material at these two links : Differential Equations 1 and Differential Equations 2.

(2) I am using Terrence Tao's book to study Analysis I and preparing my own notes in LaTeX. I plan to follow it up by Axler's new book on Measure, Integration and Analysis here. I am quite interested to learn about these ideas/abstractions, rigorous proofs. Do you think these two books are a good sequence?

Bests,
Quasar
 
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Daniel Duffy

C++ author, trainer
Regarding ODE, Brauer and Nohel is quite good.
At some stage numerical ODE is a must.

KKOP is a great book. Every MFE student should have a copy.

Axler look great and modern but might be a bridge too far just now, possibly.
 
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Quasar Chunawala

Active Member
Thanks Daniel. What would be your suggestion then, if I want to pick up probability theory and eventually measure theoretic probability at some point in the future - build a rigorous foundation? :)
 
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bigbadwolf

Well-Known Member
What would be your suggestion then, if I want to pick up probability theory and eventually measure theoretic probability at some point in the future - build a rigorous foundation?
The Axler book covers all the measure theory you need. It's one of the best books I've seen on graduate analysis and measure theory.
 

Daniel Duffy

C++ author, trainer
The Axler book covers all the measure theory you need. It's one of the best books I've seen on graduate analysis and measure theory.
Better than the Rudins, no doubt.. BTW Axler does dedicate his book to Paul Halmos.

Don't know if Iike the photos and insets (me old skule) in Axler's book. For Functional Analysis (Banach etc.) maybe this readable book is better than the few chapters in Axler?
https://www.amazon.com/Functional-Analysis-Springer-Undergraduate-Mathematics/dp/1848000049

You see more FA being used in Machine Learning research as time goes on.
And FA "influence" >> MT I reckon. You can only do so much with MT , But measures can be embedded in a special kind of Hilbert space. (RKHS)
 
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bigbadwolf

Well-Known Member

Quasar Chunawala

Active Member
Regarding ODE, Brauer and Nohel is quite good.
At some stage numerical ODE is a must.

KKOP is a great book. Every MFE student should have a copy.

Axler look great and modern but might be a bridge too far just now, possibly.
Hi Daniel,

I don't know, how big a bridge it is to go from Rudin's PMA(Baby Rudin) and Tao's Analysis I to solving Axler's book. I am sure it will require both hard-work by understanding concepts, writing proofs and also gaining significant mathematical maturity. But, what I do know, is I've stopped dwelling on these things, and I am just hungry, to learn about it, and eventually mathematical finance.
 

Quasar Chunawala

Active Member
Hi @Daniel Duffy , @bigbadwolf -

Since I have taken a first course in linear algebra and I am new to analysis, I would like to ask - what is the difference between an algebraist and an analyst? It would be nice to know from you folks. :)
 

Daniel Duffy

C++ author, trainer
Hi Daniel,

I don't know, how big a bridge it is to go from Rudin's PMA(Baby Rudin) and Tao's Analysis I to solving Axler's book. I am sure it will require both hard-work by understanding concepts, writing proofs and also gaining significant mathematical maturity. But, what I do know, is I've stopped dwelling on these things, and I am just hungry, to learn about it, and eventually mathematical finance.
I used this book in 1st year. It's great

https://www.amazon.com/Vector-Spaces-Dimension-University-Mathematical/dp/0050013580
 

bigbadwolf

Well-Known Member
Hi @Daniel Duffy , @bigbadwolf -
... what is the difference between an algebraist and an analyst? It would be nice to know from you folks. :)
Limits and continuity distinguish analysis from algebra. Interestingly enough, real analytic and topological arguments occasionally figure in the proofs of theorems about algebra. The fundamental theorem of algebra comes to mind.

Don't use Rudin's PMA if you can help it. Instead try something like Pons' Real Analysis for the Undergraduate.
 

Quasar Chunawala

Active Member
Limits and continuity distinguish analysis from algebra. Interestingly enough, real analytic and topological arguments occasionally figure in the proofs of theorems about algebra. The fundamental theorem of algebra comes to mind.

Don't use Rudin's PMA if you can help it. Instead try something like Pons' Real Analysis for the Undergraduate.
I checked out Pons book - looks to be a very clear style of exposition, a modern introductory real analysis text. In fact, I will purchase a copy of it. Two suggestions that that stood out to me from the author(I consider these gems) -

Proofs should consist of complete sentences with proper punctuation and grammar. And every problem, even if it is simply a computational exercise should be written in this fashion. Do not abuse symbology.

I am a believer that repetition is a key pedagogical technique. Let me repeat, I believe in repetition. The only way is to do and not just once. You should commit the definitions and major statements to memory.


Also, I liked this other book, it might come in handy for perhaps a hard proof - Writing proofs in Analysis, by Jonathan Kane.
 
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