# Looking for a rigorous analysis textbook

• Analysis
Homework Helper
2019 Award

## Main Question or Discussion Point

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.

## Answers and Replies

Related Science and Math Textbooks News on Phys.org
fresh_42
Mentor
If you are looking for rigor, then Bourbaki automatically comes to mind:
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.

Homework Helper
2019 Award
If you are looking for rigor, then Bourbaki automatically comes to mind:
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.

fresh_42
Mentor
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.

Last edited:
mathwonk