Analysis Readability of Rudin's Real and Complex Analysis

[bacte2013]

So I decide to self-study the real analysis (measure theory, Banach space, etc.). Surprisingly, I found that Rudin-RCA is quite readable; it is less terse than his PMA. Although the required text for my introductory analysis course was PMA, I mostly studied from Hairer/Wanner's Analysis by Its History (I did not like PMA that much). Although I said readable, I do not know if I actually understand whole materials as I am middle of first chapter, and I already have topology background from Singer/Thorpe and Engelking (currently reading). I actually like Rudin-RCA, but I am not sure if I am taking great risk as many experience people seem to not liking Rudin for learning...

Is Rudin-RCA suitable for a first introduction to the real analysis? Is it outdated? What should I know if I decide to study Rudin-RCA?

I am not planning to read the chapters in complex analysis as I am reading Barry Simon's excellent books in the complex analysis.

 

[micromass]

You know Barry Simon has other volumes on analysis too right, covering real and harmonic analysis.

But anyway, if you like Rudin, then read Rudin. But in my opinion, he's raping analysis.

 

[bacte2013]

You know Barry Simon has other volumes on analysis too right, covering real and harmonic analysis.

But anyway, if you like Rudin, then read Rudin. But in my opinion, he's raping analysis.

Yes, I actually read some pages of Simon's Part 1. However, I am worried that he starts with discussions on the Hilbert space and Fourier series first, followed by the measure theory. I thought that measure theory is used to explain them. Also, his discussions on the Borel measure and measurable functions are very different from Rudin.

By the way, why do you think Rudin-RCA is horrible for real analysis? I agree with his PMA book, but his RCA is motivating and thorough (at least from his Chapter 1).

 

[micromass]

Yes, I actually read some pages of Simon's Part 1. However, I am worried that he starts with discussions on the Hilbert space and Fourier series first, followed by the measure theory. I thought that measure theory is used to explain them. Also, his discussions on the Borel measure and measurable functions are very different from Rudin.

You can perfectly do Hilbert spaces and Fourier theory before measure theory. This is what happened historically. It forms a good motivation for measure theory. I think Simon's treatment of measure theory is superior to Rudin, but that's up to you.

 

[bacte2013]

You can perfectly do Hilbert spaces and Fourier theory before measure theory. This is what happened historically. It forms a good motivation for measure theory. I think Simon's treatment of measure theory is superior to Rudin, but that's up to you.

Could you mention specific sections that I can feel the superiority of Simon? I would like to read them and compare them with Rudin. The book by Simon is all over...

 

[bacte2013]

I actually found Barry Simon's Part-1 to be better than Rudin-RCA. He offers many integrating approaches and insights to the real analysis, just like Part-2.
I also got Kolmogorov/Fomin to supplement the Banach/Hilbert spaces, and Halmos' Measure Theory for supplement as well.