What are the best introductory books for real analysis?

[bacte2013]

Dear Physics Forum personnel,

I am a college student with huge enthusiasm to the analysis and theoretical computer science. In order to start my journey to the real analysis. I am currently taking an introductory-analysis course (Rudin-PMA; I also use Shilov too) and linear algebra (Friedberg, but I mostly use Axler) courses, and my plan is to start studying for the real analysis and functional analysis right after the end of final exams. I have been looking for the good introductory books on the real analysis (as for the functional analysis, I bought Kreyszig), but there are just too many of them to select the few good ones...My future plan is to read Rudin-RCA during Summer of 2016, so my foundational plan is to use the Winter break and Spring Semester to read the introductory real-analysis book.

I visited the mathematics library and went through a collection of books, and I did like Kolmogorov/Fomin, Carothers, and Stein/Sharkachi, but I am not sure if any of them is good for the pedagogical learning since I only read the few pages of the first chapters in those books. Could you inform me if any of them is good for the first introduction to the real analysis, and/or inform me some other books for the learning?

Beside from Rudin, Shilov, and Axler, I read the topology sections of Simmons' "Introduction to Topology and Modern Analysis". Do I need additional background in the topology?

 

[micromass]

What subjects are you interested in covering?

 

[bacte2013]

What subjects are you interested in covering?

The measure theory, integration theory, and Hilbert's Space.

 

[micromass]

The measure theory, integration theory, and Hilbert's Space.

Hilbert space is functional analysis.

Anyway, if you are only interested in measure and integration theory, then books like Carothers are not what you're looking for. I recommend the following books:

1) Bartle "The elements of integration and Lebesgue measure" This is a quite elementary but very good book. It contains everything the usual analyst should know about measure theory.

If you are somewhat interested in other topics, then the following books are good too:

2) Lang's "real and functional analysis" (do not confuse this with his "undergraduate analysis"). This book does measure theory and analysis in the very general setting of Banach spaces. This is overkill for most people, but I found the book very exciting.

3) Conway's "A course in abstract analysis". This contains a high-level introduction to measure theory and integration. It is not elementary at all, but it is a very nice approach (which is basically constructing the Lebesgue measure and integral using the Daniell integral approach). It then continues with functional analysis over Hilbert spaces, Banach spaces and topological vector spaces.

4) Billingsley's "Probability and measure" This was my introduction to measure theory. It does it in the context of probability theory, but it is a really well-written book even if you are mostly interested in analysis. It has extremely good exercises. If you are somewhat interested in probability theory, then this book is a must.

 

[bacte2013]

Hilbert space is functional analysis.

Anyway, if you are only interested in measure and integration theory, then books like Carothers are not what you're looking for. I recommend the following books:

1) Bartle "The elements of integration and Lebesgue measure" This is a quite elementary but very good book. It contains everything the usual analyst should know about measure theory.

If you are somewhat interested in other topics, then the following books are good too:

2) Lang's "real and functional analysis" (do not confuse this with his "undergraduate analysis"). This book does measure theory and analysis in the very general setting of Banach spaces. This is overkill for most people, but I found the book very exciting.

3) Conway's "A course in abstract analysis". This contains a high-level introduction to measure theory and integration. It is not elementary at all, but it is a very nice approach (which is basically constructing the Lebesgue measure and integral using the Daniell integral approach). It then continues with functional analysis over Hilbert spaces, Banach spaces and topological vector spaces.

4) Billingsley's "Probability and measure" This was my introduction to measure theory. It does it in the context of probability theory, but it is a really well-written book even if you are mostly interested in analysis. It has extremely good exercises. If you are somewhat interested in probability theory, then this book is a must.

Thank you very much! I need to check out Bartle, Conway, and Billingsley! About Lang's book, I am not quite comfortable with his style as I personally feel that his prose is incomplete and written in hurried way...But I should definitely check it out. Oh! I actually got a free copy of Stein/Sharkachi from my professor. Should I read that after reading one of the books you mentioned?

 

[micromass]

Thank you very much! I need to check out Bartle, Conway, and Billingsley! About Lang's book, I am not quite comfortable with his style as I personally feel that his prose is incomplete and written in hurried way...But I should definitely check it out. Oh! I actually got a free copy of Stein/Sharkachi from my professor. Should I read that after reading one of the books you mentioned?

Stein & Shakarchi is of course a very good book. You could read that as an introduction too. The only problem is that you are looking for Volume 3 of the series. So you'll need to read the previous 2 volumes too (although I don't think you'll miss much).

 

[bacte2013]

Stein & Shakarchi is of course a very good book. You could read that as an introduction too. The only problem is that you are looking for Volume 3 of the series. So you'll need to read the previous 2 volumes too (although I don't think you'll miss much).

Thank you very much for the advice. I decided to read both Bartle and Kolmogorov/Formin to study the real analysis, then proceed to Stein/Shakarchi, and study either Lang or Conway. I think it would be beneficial for me to learn the basics first from Bartle and K/F (especially K/F) before proceeding to other books (I got an impression that both Lang and Conway expect the prospective readers a basic understanding of the structures of R^n and measure theory).

As for the functional analysis, is either Shilov or Kreyszig good as a starting point? I am also confused about K/F as they wrote both real-analysis and functional-analysis books but both of them cover the same materials.

 

[Joshua L]

The Elements of Real Analysis, Second Edition 2nd Edition by Robert G. Bartle.

http://www.amazon.com/gp/product/047105464X/?tag=pfamazon01-20