Which book will suit the following course syllabus (introductory analysis)?

[bacte2013]

Dear Physics Forum personnel,
I am a undergraduate student with math and CS major who is currently taking an introductory analysis course called MATH 521 (Rudin-PMA). On the next semester, I will be taking the course called MATH 522, which is a sequel to 521. My impression is that 522 will be an analysis on manifolds, so I have been browsing books like Spivak, Munkres, Hubbard/Hubbard, Fleming, etc. However, I learned today that the course outline of 522 is deviated from my initial impression of it being the multivariable-analysis course. According to the course outline (URL is below), it seems the topics touch more or less the real analysis and topology. The official textbook is Rudin-PMA, but I do not think that book covers many topics for the 522. Could you suggest me some books dedicated to 522?

URL: https://www.math.wisc.edu/sites/default/files/521-522_0_1.pdf (very last page)
URL: http://www.math.wisc.edu/~beichman/Syllabus522F14.pdf (slight deviation)
URL: https://www.math.wisc.edu/~seeger/522/syl.pdf (another deviated syllabus)

Should I get the books on real and functional analysis like Rudin-RCA, Stein/Sharkachi, Kolmogorov, Simmons, Lang-RFA?

 

[mathwonk]

you already seem to have plenty of books. it appears you will likely need to study the contraction principle in complete metricm spaces, and its application to the implict function theorem in a normned space, and perhaps the existence theorem for ordinary differential equations. so i would suggest looking in your books for that topic. as i recall, neither spivak nor fleming use that approach to the inverse and implicit function theorems in their advanced calculus books, but lang does in his book Analysis I (1968), later published under a different title, as does spivak in his book on differential geometry, vol. 1, at least for the diff eq theorem.

the point is perhaps that one does not need the contraction lemma approach to the inverse function theorem in finite dimensions, but it is standard in infinite dimensional banach spaces.

 

[bacte2013]

you already seem to have plenty of books. it appears you will likely need to study the contraction principle in complete metricm spaces, and its application to the implict function theorem in a normned space, and perhaps the existence theorem for ordinary differential equations. so i would suggest looking in your books for that topic. as i recall, neither spivak nor fleming use that approach to the inverse and implicit function theorems in their advanced calculus books, but lang does in his book Analysis I (1968), later published under a different title, as does spivak in his book on differential geometry, vol. 1, at least for the diff eq theorem.

the point is perhaps that one does not need the contraction lemma approach to the inverse function theorem in finite dimensions, but it is standard in infinite dimensional banach spaces.

Do you mean Lang's Real and Functional Analysis or Undergraduate Analysis? For latter, I do not recall the treatment of normed space.

 

[mathwonk]

The book I named has been reissued as Undergraduate Analysis, with only a few changes. Differential calculus in normed banach spaces is treated there in chapters Vi, VII, VIII, and XVI, XVII, and XVIII. as you can see on amazon.

 

[bacte2013]

How about Loomis/Sternberg? I believe you mentioned that it also does the differentiation and integration on the Banach space. Honestly, I do not like Lang's books in general.

 

[mathwonk]

Forgive me, but I am beginning to feel you may be wasting time with these questions, mostly your own time. I suggest you need to get to reading and thinking and quit "dancing around the fire". Yes Loomis and Sternberg covers the contraction mapping approach, but that book is not really recommended for learning, and I suspect if you will look at them, you may find Lang's book easier to learn from. But you should be the judge. I can say however that Lang's many books differ from each other greatly in accessibility, and you cannot judge his undergraduate analysis by his graduate analysis or his differential manifolds, or his algebra book. I think I have said about all I have to say on this. good luck.