What are the best introductory books for real analysis?

In summary, the college student is seeking advice on introductory books for real analysis and functional analysis. They have already read introductory analysis and linear algebra courses and plan to start studying for real analysis and functional analysis. They are looking for good introductory books on real analysis and have narrowed it down to Kolmogorov/Fomin, Carothers, and Stein/Sharkachi, but are unsure of which one to choose. They also mention reading the topology sections of Simmons' "Introduction to Topology and Modern Analysis" and ask if they need additional background in topology. Their main interests are in measure theory, integration theory, and Hilbert's space. They are recommended Bartle, Lang, Conway, and Billingsley's books for study, with
  • #1
bacte2013
398
47
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?
 
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  • #2
What subjects are you interested in covering?
 
  • #3
micromass said:
What subjects are you interested in covering?

The measure theory, integration theory, and Hilbert's Space.
 
  • #4
bacte2013 said:
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.
 
  • #5
micromass said:
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?
 
  • #6
bacte2013 said:
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).
 
  • #7
micromass said:
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.
 

Related to What are the best introductory books for real analysis?

1. What is "Real Analysis"?

"Real Analysis" is a branch of mathematics that deals with the study of real numbers and their properties, including limits, continuity, differentiation, and integration.

2. What are some good books for learning Real Analysis?

Some popular and highly recommended books for learning Real Analysis include "Principles of Mathematical Analysis" by Walter Rudin, "Introduction to Real Analysis" by Robert G. Bartle and Donald R. Sherbert, and "Real Mathematical Analysis" by Charles Pugh.

3. Is prior knowledge of calculus required for studying Real Analysis?

Yes, a strong understanding of calculus is necessary for studying Real Analysis. In particular, knowledge of limits, continuity, and differentiation is essential.

4. How can I apply Real Analysis in real life situations?

Real Analysis has numerous applications in fields such as physics, engineering, economics, and statistics. It can be used to model and analyze real world phenomena, make predictions, and solve complex problems.

5. Are there any online resources or tutorials available for learning Real Analysis?

Yes, there are many online resources and tutorials available for learning Real Analysis. Some popular options include MIT OpenCourseWare, Khan Academy, and YouTube channels such as TheMathStudent and Dr Chris Tisdell.

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