Okay I won't bother responding to the complaints regarding several variable theory since I didn't study that from Rudin. Responding respectively,
4) Rudin does not neglect series developments. Also, Laurent series are actually left to the exercises since it probably detracts from his particular treatment of basic complex variables in a single chapter.
2) What complicated algorithm? I'm pretty sure he just calls the winding number the index and then uses it to give mostly standard proofs. Not sure what your complaint is here.
3) Eh, I felt the analogy with basic topology helped in getting down the basics of measurable sets and measurable functions. Obviously general topology and analysis begin to differ when you start talking about say countability axioms and metrization problems, but the similarities between basic facts regarding continuous functions and those of measurable functions are apparent to a lot of people, including myself.
4) Please explain why it is not nicely done in Rudin instead of suggesting alternative constructions. Why should I follow a daniell approach?
5) This is somewhat of a legit complaint. Yes Rudin does not always give context for the many theorems he puts together, but this is exactly why it also makes a good reference. I don't think his presentation really affects his clarity and exposition skills though.
6) You're thinking of his wife, who studied under Moore. Walter Rudin studied under John Jay Gergen at Duke. In any case, does studying math under a racist make you a racist?
6) Eh I disagree, but then again I mostly stayed away from his texts while studying functional and fourier on LCA groups.
Anyways, the advantages of Rudin are myriad, the disadvantages are trivial, and further argument will lead to loss of valuable analysis time.
