Analysis Looking for a rigorous analysis textbook

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1. Jun 22, 2017

Math_QED

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.

2. Jun 22, 2017

Staff: 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.

3. Jun 22, 2017

Math_QED

My university uses self made books, so that wouldn't be an option unfortunately.

4. Jun 22, 2017

Staff: Mentor

http://www.cmls.polytechnique.fr/perso/laszlo/webTC/polymat311.pdf

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: Jun 22, 2017
5. Jun 23, 2017

mathwonk

i recommend spivak's calculus.

6. Jun 26, 2017

MidgetDwarf

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.

7. Jun 30, 2017

Tukhara

Barry Simon's textbook series: A Comprehensive Course in Analysis. Amazing; it's probably one of my favorites if not my most loved.