Came across an Analysis post on this board while searching the net

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Hi All,

I was searching for Analysis course that typically follows Spivak's Calculus.

While searching the Net came across this post below and have couple questions if you don't mind:

In the old days, the progression was roughly: rigorous one variable (Spivak) calculus, Abstract algebra (Birkhoff and Maclane), rigorous advanced calculus (Loomis and Sternberg), introductory real and complex analysis via metric spaces as in Mackey's complex analysis book, general analysis as in Royden, (big) Rudin, or Halmos and Ahlfors, algebra as in Lang, and algebraic topology as in Spanier. Then you specialize.

In particular Spivak was written for a first semester freshman book.

My questions:

1. "Rigorous Advanced Calculus (Loomis and Sternberg)". Is that a single book or two different books by different authors? Couldnt find it on Amazon, but found Advanced Calculus by Loomis. What is the relation of the second book to the suggested one? Are they interchangeable? If not, what's the best alternative?

2. "introductory real and complex analysis via metric spaces as in Mackey's complex analysis book". I couldn't find it anywhere. What would be the best alternative?

3. "general analysis as in Royden". Found Real Analysis (4th Edition) by
Halsey Royden, Patrick Fitzpatrick. Are these the same books? Or are they interchangeable? If not, what would be the modern alternative to the suggested text?

4. "(big) Rudin, or Halmos and Ahlfors". I apologize for my ignorance, but could someone, please, elaborate on this one? Couldnt locate the book written by both Halmos and Ahlfors together.

Thank You.

 

[jbunniii]

1. "Rigorous Advanced Calculus (Loomis and Sternberg)". Is that a single book or two different books by different authors? Couldnt find it on Amazon, but found https://www.amazon.com/dp/0867201223/?tag=pfamazon01-20. What is the relation of the second book to the suggested one? Are they interchangeable? If not, what's the best alternative?

Yes, the book at your link is the right one. The title is Advanced Calculus, and it is a (very) rigorous book. It is out of print, and I see that the used copies on Amazon are obscenely priced. Fortunately, Sternberg has made it available for free in PDF form at his web site: http://www.math.harvard.edu/~shlomo/

3. "general analysis as in Royden". Found https://www.amazon.com/dp/013143747X/?tag=pfamazon01-20 Real Analysis (4th Edition) by
Halsey Royden, Patrick Fitzpatrick[/url]. Are these the same books? Or are they interchangeable? If not, what would be the modern alternative to the suggested text?

Yes, that's the right book. The earlier editions had Royden as the sole author, but he died in the 1990s. The 4th edition was updated by another author, Fitzpatrick. Be careful about the new edition: there are people complaining even in the past few days that the text contains Latex errors:

This book is clearly damaged. The publisher is selling copies of the book that cannot be used. References to equations, theorems, references, are often numbered with two question marks (??) rather than giving the correct number. This is an elementary error. The book was typeset in Latex, and it wasn't compiled correctly. Most Latex users are familiar with this error, and know how to fix it. Apparently, this publisher is not.

4. "(big) Rudin, or Halmos and Ahlfors". I apologize for my ignorance, but could someone, please, elaborate on this one? Couldnt locate the book written by both Halmos and Ahlfors together.

Thank You.

Big Rudin: https://www.amazon.com/dp/0070542341/?tag=pfamazon01-20
Halmos: https://www.amazon.com/dp/0387900888/?tag=pfamazon01-20
Ahlfors: https://www.amazon.com/dp/0070006571/?tag=pfamazon01-20

Big Rudin covers both real and complex analysis. Halmos is real analysis, Ahlfors is complex analysis. So mathwonk presumably meant, get either Big Rudin, or get both Halmos and Ahlfors.

 

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Awesome! I appreciate you took your time to clarify all this. Thank You, Jbunniii.