Recommended texts for self-study [real & functional analysis]

In summary, a basic knowledge of real and functional analysis would be helpful for self-study, but Rudin is not accessible to people without prior background in the topic. Wade and Gallian are good resources for refresher courses, and Complex Analysis and Functional Analysis can be studied in tandem.
  • #1

boneh3ad

Science Advisor
Insights Author
Gold Member
3,441
1,231
Hello,

I have been increasingly running into topics in my field where at least a basic faculty with real and functional analysis would be quite helpful and I would like to go about self-studying a bit in that area. I know that Rudin is the canonical text in the field, but I have also heard that it is not all that accessible for people who are coming into the field with no prior background in the topic. I was wondering if some of you guys might give me a sense of a nice way to approach the topic.

My background is okay, I guess. I have seen the ε-δ definition of a limit, for example, but it doesn't go much deeper than that into analysis. I was thinking about grabbing Spivak's Calculus as a rigorous refresher of sorts and then moving on to Rudin and/or Carothers for real analysis and then possibly Kreyszig's Intorductory Functional Analysis with Applications for that topic. Does this seem like a decent plan of attack or do any of you have any other suggestions?

Thanks.
 
Physics news on Phys.org
  • #2
I used Wade for Real Analysis in undergrad. It's good. It has everything you need to refresh on from calculus, even vector calculus, and its really readable. It has everything you need on convergence of sequences and functions, differentiability and integrability some Fourier analysis and even topology in the reals. Right now I'm using Kreyszig for a graduate class. I've also done "Real Analysis: Measure Theory Integration and Hilbert Spaces", which is more complete than Kreysig but not as approachable as Wade. You want to make sure your proof writing is pretty solid before doing any of these, first doing to first half or so of "A Transition to Advanced Mathematics" by Smith, Eggen, and St. Andre.

For complex I've used "Complex Analysis" by Bak and Newman and "Complex Functiom Theory" by Sarason. The latter is very pretty approachable. You should save Rudin for later.

I would recommend that you also do some Algebra while you work through Wade. The difference in the process of the proofs will be of benefit. "Contemporary Abstract Algebra" by Gallian is incredibly easy to read and has a ton of exercises. I've used "Abstract Algebra" by Dummit And Foote also, it's a bit more in depth than Gallian but Gallian's writing style and format are great. You don't need to do linear algebra before abstract. I finished Gallian before taking Linear Algebra and I thought already having a good abstract foundation made my study of Vector Spaces far richer.

If you do the transition book, Wade, Gallian, and Sarason you'll be on equal footing with most 4th year Math undergrads.

You can PM me if you have any more specific questions.
 
Last edited:
  • Like
Likes 1 person
  • #4
I appreciate the feedback from both of you.
 
  • #6
A David said:
For complex I've used "Complex Analysis" by Bak and Newman and "Complex Function Theory" by Sarason. The latter is very pretty approachable. You should save Rudin for later.

I should note that Sarason is seriously lacking in exercises.
 
  • #7
Kreyszig is the minimal starting point for Functional Analysis. I am not saying it's bad, but it's very lightweight. I would call it a prerequisite to start studying functional analysis. If you can afford only one book of that kind, I'd go with Debnath and Mikusinski's "Introduction to Hilber Spaces with Applications".
https://www.amazon.com/dp/0122084381/?tag=pfamazon01-20
 

1. What is the purpose of studying real and functional analysis?

The purpose of studying real and functional analysis is to gain a deeper understanding of mathematical concepts and techniques that are fundamental to advanced mathematics. This includes topics such as limits, continuity, differentiation, integration, and more.

2. What are some recommended texts for self-study in real and functional analysis?

Some recommended texts for self-study in real and functional analysis include "Principles of Mathematical Analysis" by Walter Rudin, "Real Analysis" by Royden and Fitzpatrick, and "Functional Analysis" by Walter Rudin.

3. How do I approach self-study in real and functional analysis?

It is important to have a strong foundation in basic calculus and linear algebra before diving into real and functional analysis. It is also helpful to have a study plan and to work through exercises and problems in the recommended texts. Seeking help from online resources or a tutor can also be beneficial.

4. What are some common challenges faced when studying real and functional analysis?

Some common challenges when studying real and functional analysis include understanding the abstract concepts and theorems, mastering the rigorous proofs, and applying the concepts to solve problems. It is important to practice regularly and seek help when needed.

5. How can studying real and functional analysis benefit my career as a scientist?

Studying real and functional analysis can benefit a scientist's career by providing a solid foundation in advanced mathematical concepts and techniques. This can be helpful in conducting research, analyzing data, and developing models and theories. It can also open up opportunities for collaboration with mathematicians and other scientists in related fields.

Suggested for: Recommended texts for self-study [real & functional analysis]

Replies
13
Views
2K
Replies
2
Views
788
Replies
14
Views
1K
Replies
3
Views
2K
Replies
1
Views
921
Replies
16
Views
2K
Replies
13
Views
2K
Back
Top