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I am searching for a real analysis book (instead of Rudin's) which satisfies the following requirements:

  • clear, motivated (but not chatty), clean exposition in definition-theorem-proof style;
  • complete (and possibly elegant and explicative) proofs of every theorem;
  • examples and solved exercises;
  • possibly, the proofs of the theorems on limits of functions should not use series;
  • generalizations of theorem often given for $\mathbb{R}$ to metric spaces and also to topological spaces.

Thank you very much in advance for your assistance.

user62029
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    Related: http://mathoverflow.net/questions/10282/alternative-undergraduate-analysis-texts –  Dec 07 '14 at 05:30

11 Answers11

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Analysis 1 and Analysis 2 by Terence Tao.

  • I will be interested to hear your objections against this book @GeorgeChen. –  Feb 07 '18 at 07:10
  • Incidentally, I was hoping to discuss with you regarding some clarification of Russell's comments. I will be glad if you come to the Philosophy of Mathematics room. –  Feb 07 '18 at 07:15
  • Words like counting, forward, indefinitely, equal, different, ++, true, 4, 6, object, unordered, etc. are used before they were defined. He counted your "intuition" to understand his "informal" definitions not realizing that a precise definition is the first step to dwell upon a concept. This habit of mind betrays blatant disregard for meanings. This book is childlike, in the sense that it relies heavily on make believe. – George Chen Feb 09 '18 at 16:08
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    At page 15, he specifically asked the readers to perform a rather difficult task: try to set aside, for the moment, everything you know how to count, to add, ... Yet, it is impossible to understand his definitions and axioms without borrowing from what we already know. Specifically, where was equal defined? – George Chen Feb 10 '18 at 03:03
  • A good example of his muddleheadedness is his proof that 0.5 is not a natural number in the 3rd paragraph on page 22. In Peano's axiom, ++ means "successor of," not necessarily +1; in Tao's proof, ++ is interpreted as +1 before addition is defined. As a matter of fact, axiom 2.5 (principle of mathematical induction) cannot rule out 0.5 from the set of natural numbers if we do not take the liberty of interpreting "successor of" as +1. – George Chen Feb 10 '18 at 03:16
  • W & R had showed that any progression suffices to make Peano's axioms true. If a progression of numbers, such as 0, 0.5, 1, 1.5 ,2, 2.5 ..., is laid down one number after another, Peano's axioms will still be true, where 0.5 is the successor of 0, 1 the successor of 0.5, etc. – George Chen Feb 10 '18 at 03:34
  • And mathematical induction will still be true if we substitute "successor of" for +1. In other words, if a property is hereditary with respect to "successor of" and is possessed by 0, it is possessed by all members of this progression. – George Chen Feb 10 '18 at 03:57
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    Regarding your second and third comments see this post @GeorgeChen. –  Feb 10 '18 at 04:16
  • Also note that the equality axioms were defined probably in one of the Appendixes. –  Feb 10 '18 at 04:17
  • Also @GeorgeChen, regarding the clarification of Russell's comments which I mentioned here, I have now posted a question here. –  Feb 10 '18 at 04:28
8

Mathematical Analysis by Tom M Apostol

Mr. Math
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I recommend a combination of books Real Mathematical Analysis (Undergraduate Texts in Mathematics) by Charles C. Pugh together with Elementary Classical Analysis by Jerrold E. Marsden, Michael J. Hoffman.These books are concise, motivate the theorems are an elegant presentation.But do not introduce topological spaces.

I believe the book that will satisfy all the requirements of the question will be: Analysis I and Analysis II written by Vladimir A. Zorich. This book has the disadvantage of having an encyclopedic character.

The Zorich's book brings the generalizations of theorem is often do given to $\mathbb{R}$ to metric spaces and topological spaces also to.

Elias Costa
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Lang's "Real and Functional Analysis" is very thorough and to the point. Rather than presenting theorems for $\mathbb{R}$ and then generalizing to metric/topological spaces, Lang typically presents theorems in their most general form up front and then specializes as necessary to illustrate important cases. This is, in fact, my favorite part of this book: Lang has some extremely general (and elegant) constructions that I have not seen anywhere else. The prime example here is the Lebesgue integral, which usually is developed first for positive, real-valued functions, then extended to general real-valued functions, then to vector-valued functions, etc. Instead, Lang jumps straight to integrating functions on Banach spaces, knocking off all the aforementioned cases simultaneously. This results in a much more streamlined presentation (in my opinion).

The price you have to pay for the greater generality is some loss of motivation. It's worth noting in this vein that many analysists are not fond of Lang's book (presumably for this reason). I think it's worthwhile (especially if you intend to use this as a supplement to books like Rudin's).

Sesquipedal
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5

Intro to Real Analysis by Brannan from Cambridge University Press.

DeepSea
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The Elements of Real Analysis by Robert Bartle

Methods of Real Analysis by Goldberg (harder to find maybe)

Foundations of Modern Analysis by Friedman (perhaps more terse than what you are after but excellent at highlighting what's important)

JohnD
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  • Could you tell me the difference between the two analysis books by Bartle, the other one being Introduction to Real Analysis? – StubbornAtom Jan 22 '17 at 14:19
  • @StubbornAtom I think Intro to Real Analysis only covers the real line, whereas Elements covers R^n. – JohnD Jan 24 '17 at 16:34
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Real Analysis by Royden and Fitzpatrick...

...+ solutions manual

BCLC
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When I asked this question to my professor he told me to get 'Introduction to Analysis' by Maxwell Rosenlicht. It's a short read, but It covered everything I needed for the course.

Fmonkey2001
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I really like Howland's 'Basic Real Analysis'. The only unfortunate bit is that the topology proofs are in appendices. That said, the book is readable with only basic calculus, and really well-written for the student (rather than as a reference text).

user102830
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1

Real Analysis for Graduate Students by Richard Bass.

Robert Cardona
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Transcript of Vaughan Jones's Lecture Notes (virtually verbatim)

Mike Pierce
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