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

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boneh3ad
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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.
 

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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.
 
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boneh3ad
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I appreciate the feedback from both of you.
 
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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.
 
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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
 

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