Analysis Texts

Analysis is typically considered one of the most complicated subjects in mathematics.  The subject is certainly very challenging to study on ones own and a good professor is typically very necessary.  However, from what I've read in Analysis I have yet to encounter a decent introductory text on the subject that is motivated for students.  Many texts appear illusive, to me, in their explanations and are not really illuminating.  That being said, the big names in classic analysis texts are often extremely terse or dense, this makes them practically impenetrable for a beginner.  However, some gentler introductions are referenced below, though they may not be the best.  I am hoping my tutorials will supplement some of the lacking texts out there.

"Mathematical Proofs: A Transition to Advanced Mathematics" by Gary Chartrand, Albert D. Polimeni, and Ping Zhang.

Pages: 365
ISBN: 978-0-321-39053-0
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"An Introduction to Analysis" by Gerald G. Bilodeau, Paul R. Thie, and G.E. Keough.

Of the Analysis books I've looked at this had the most gentle introduction to the topic.  However, I still found the book relatively mystifying at parts.  My class on Real Analysis used this as the text, but without the help of a professor I am not sure how much this book could ever stand on it's own.  Like most of these texts the book provides you with proofs that are complete and a little exposition, however it does not fully explain in detail the thought process behind doing the proofs.  This is the books one failing.

Pages: 333
ISBN: 978-0-7637-7492-9
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"Calculus" by Michael Spivak.

Do not be fooled by the title of this book, this is surely a text dealing with Real Analysis.  Spivak is probably one of the finest authors of mathematics out there and I quite like his books.  The only problem I can see for trying to get through this text is that his problems in the exercises can quickly become impossible to get through.  However, if you can manage the problems, then you are probably quite a good thinker when it comes to mathematics.

Pages:  680
ISBN: 978-0-914098-91-1
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"Principles of Mathematical Analysis" by Walter Rudin.

This is a classic book in Analysis, but it is extremely difficult going if you are a beginner.  Despite what anyone claims about it being "introductory", I would not recommend this kind of a text unless one is already familiar with how this kind of material works.  Lest you find yourself hopeless lost in one of Rudin's proofs one day!  However, if you can get through this text and understand it well, then you will have reasonable assurance that you have understood the basics of analysis extremely well.  A must read for moving on to much higher level works.

Pages: 342
ISBN: 978-007-054235-8
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"Real & Complex Analysis" by Walter Rudin.

Pages: 416
ISBN: 978-007-054234-1
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"Topology" by James R. Munkres.

Pages: 537
ISBN: 0-13-181629-2
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 "Analysis on Manifolds" by James R. Munkres.

Pages: 366
ISBN: 0-201-51035-9
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 "Differential Geometry of Curves and Surfaces" by Manfredo P. do Carmo.

Pages: 501
ISBN: 0-13-212589-7
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"Riemannian Geometry" by Manfredo P. do Carmo.

Pages: 300
ISBN: 978-0-8176-3490-2
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