Looking for a rigorous analysis textbook

In summary, the conversation involved a mathematics undergrad student seeking a book on rigorous real analysis, with a focus on topics such as construction of ##\mathbb{R}##, limits, sequences, derivatives, series, and integrals. Recommendations were given for books such as Bourbaki, Hewitt, Stromberg, and Shilov, but ultimately the student was recommended to try Spivak's Calculus or Barry Simon's A Comprehensive Course in Analysis. It was also suggested to visit a bookstore to find a standard book for real analysis and to prepare a list of desired topics.
  • #1
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Hello all. I'm a mathematics undergrad student who finished his first university year succesfully. I got courses of calculus, but these weren't very rigorous. I did learn about stuff like epsilon and delta proofs but we never made exercises on those things. The theory I saw contained proofs but the main goal of the course was to succesfully learn to solve integrals (line integrals, surface integrals, double integrals, volume integrals, ...), solve differential equations, etc.

I already took proof based courses like linear algebra and group theory, so I think I am ready to start to learn rigorous real analysis, so I'm looking for a book that suits me.

I want the book to contain the following topics:

The usual analysis stuff:

- a construction of ##\mathbb{R}## or a system that takes ##\mathbb{R}## axiomatically for granted
- rigorous treatment of limits, sequences, derivatives, series, integrals
- the book can be about single variable analysis, but this is no requirement
- exercises to practice (I want certainly be able to prove things using epsilon and delta definitions after reading and working through the book)

Other requirements:

- The book must be suited for self study (I have 3 months until the next school year starts, and I want to be able to prepare for the analysis courses).

I have heard about the books 'Real numbers and real analysis' by Ethan D. Block and 'Principles of mathematical analysis' by Walter Rudin, and those seem to be good books. I have also heard these books are very hard to start with, so maybe I need something easier to start with.

Can someone hint me towards a good book? If you want me to add information, feel free to leave a comment.
 
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  • #2
If you are looking for rigor, then Bourbaki automatically comes to mind:
https://www.physicsforums.com/threads/what-is-a-tensor-comments.917927/#post-5788263
However, you should have a look beforehand as it is not everybody's taste, the way they present mathematics. And it might be quite a few instead of one. Shouldn't it be the best to follow the books used at your university(-ies)? Personally I only know those used at my university which are rather basic (and in the wrong language), or one from Hewitt, Stromberg, which might be a bit too abstract, as it is centered around measure algebras.
 
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  • #3
fresh_42 said:
If you are looking for rigor, then Bourbaki automatically comes to mind:
https://www.physicsforums.com/threads/what-is-a-tensor-comments.917927/#post-5788263
However, you should have a look beforehand as it is not everybody's taste, the way they present mathematics. And it might be quite a few instead of one. Shouldn't it be the best to follow the books used at your university(-ies)? Personally I only know those used at my university which are rather basic (and in the wrong language), or one from Hewitt, Stromberg, which might be a bit too abstract, as it is centered around measure algebras.

My university uses self made books, so that wouldn't be an option unfortunately.
 
  • #4
How about this one:
http://www.cmls.polytechnique.fr/perso/laszlo/webTC/polymat311.pdf

Looks promising: reliable source and for free.

Another idea is to go to the bookstore where students usually buy their natural sciences and mathematics books and ask for a standard book for real analysis. I would also prepare a list of topics you definitely want to find (quickly) in those books, which might be partitioned as "Real Analysis I" (in one variable), "Real Analysis II" (in n variables), "Complex Analysis", "Differential analysis", "Integration" or similar. I'd like to make a list for you, but it would get very long. Guess that's why the Bourbaki link above contains several books.
 
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  • #5
i recommend spivak's calculus.
 
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  • #6
Maybe Shilov: Elementary Real and Complex Analysis?

If you have never gone through a rigorous math book before. It would be better to pick up something like Spivak. To get aquainted with reading and writing proofs.
 
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  • #7
Barry Simon's textbook series: A Comprehensive Course in Analysis. Amazing; it's probably one of my favorites if not my most loved.
 
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1. What is a rigorous analysis textbook?

A rigorous analysis textbook is a resource used in mathematics and science fields that focuses on the logical and theoretical foundations of mathematical concepts and proofs. It is typically used in advanced courses and is meant to challenge students' critical thinking skills.

2. What should I look for in a rigorous analysis textbook?

When looking for a rigorous analysis textbook, it is important to consider the level of rigor and depth of the material, the clarity of explanations and examples, and the relevance of the topics to your field of study.

3. How do I know if a textbook is rigorous enough?

A textbook can be considered rigorous if it presents concepts and proofs in a logical and systematic manner, provides challenging exercises and problems, and encourages critical thinking and understanding rather than memorization.

4. Can a rigorous analysis textbook be used for self-study?

Yes, a rigorous analysis textbook can be used for self-study, but it is recommended to have a background in mathematics and some experience with proofs before attempting to study from it independently.

5. Are there any recommended textbooks for rigorous analysis?

There are many rigorous analysis textbooks available, and the best one for you may depend on your specific field of study and learning style. Some popular options include "Principles of Mathematical Analysis" by Walter Rudin, "Real Analysis" by Royden and Fitzpatrick, and "Introduction to Analysis" by Maxwell Rosenlicht.

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