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I was asked if I could suggest an "honest" introductory real analysis book, where "honest" means:

  • with every single theorem proved (that is, no "left to the reader" or "you can easily see");
  • with every single problem properly solved (that is, solved in a formal (exam-like) way).

I've studied using Rudin mostly and I liked it, but it really doesn't fit the description, so I don't know what book I should suggest. Do you have any recommendations?

Update: I need to clarify that my friend has just started to study real analysis and the course starts from the very basics, deals with real valued functions of one variable, but introduces topological concepts and metric spaces too.

Dal
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    are you looking for an introduction to elementary real analysis (i.e., real valued functions of a single real variable) or an introduction to analysis a la Rudin, assuming the elementary things are known and aiming at topology, metric space theory etc.? – Ittay Weiss Nov 30 '14 at 20:56
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    It is strange, but an honest introduction to analysis is Spivak's Calculus. – Artem Nov 30 '14 at 23:40
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    I can't help but remark/claim that your friend should not really study from such a thing, etc. That is, carrying out every detail, and doing so "properly" (code for exaggerated-formal) is substantially misguided. – paul garrett Dec 01 '14 at 00:07
  • Duplicate of http://math.stackexchange.com/questions/1037380/real-analysis-book-suggestion ? – Barry Cipra Dec 01 '14 at 00:07
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    @paulgarrett Could you please elaborate on that a bit? – littleO Dec 01 '14 at 01:14
  • Exactly what real analysis topics do you expect to cover? Presumably either construction of the reals from integers or rationals, or at least an axiomatic treatment (which ought to include uniqueness up to order isomorphism); convergence of sequences and numerical series along with Taylor series and, more generally, convergence & uniform convergence of sequences and series of functions. The critical issue is which integral: just the Riemann integral? Riemann-Stieltjes? or Lebesgue? The 1st edition just did Riemann-Stieltjes integration, as I recall; the current, 3rd edition includes Lebesgue. – murray Dec 01 '14 at 04:39
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    @littleO, to invest one's time and energy in the proposed fashion is, to my mind, suboptimal in at least two ways. First, surely one has caught on to the general pattern of the low-level details after a certain number of example-proofs, without having to continue and see every other idea accompanied by all those low-level details... which tend to swamp the main idea. Second, getting into the habit of conceiving of the activity of mathematics as essentially involving writing out all possible details, rather than choosing the most-relevant, critical details, is simply bad practice. – paul garrett Dec 01 '14 at 13:56
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    How does Rudin not fit your requirements? I've not read a huge amount of it but I've recently gone through parts of it in detail and I don't remember any proofs being left to the reader or any unsolved problems (except for the exercises at the end of each chapter). – David Richerby Dec 01 '14 at 14:20
  • @DavidRicherby: I missed this the first time I read Rudin also. But now I'm pretty sure I could flip to a random page in Rudin and it would have some sort of "I'm not wasting bookspace on this" comment. I think the first instance I remember distinctly was his proof of Cauchy-Schwarz, which relies on the reader to do a considerable amount of scratchwork. (The reason I remember this distinctly is because one of my homework problems was to write clearly everything that he left out…) – Eric Stucky Dec 08 '14 at 08:07
  • @EricStucky I have a copy of Rudin with me now (3rd ed). What he calls "the Schwarz inequality" is proved as Theorem 1.35; there don't seem to be any gaps in it. – David Richerby Dec 08 '14 at 08:12
  • Apparently I am the one with memory problems; I am also looking at the theorem and don't find it objectionable. Now I need to find that homework assignment… – Eric Stucky Dec 08 '14 at 08:25
  • Does this answer your question? Good book for self study of a First Course in Real Analysis – user1110341 Oct 01 '23 at 06:57

10 Answers10

23

Possibly Abbott, Understanding Analysis

Git Gud
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Simon S
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A bit of self publicity, but the reason that A Primer on Hilbert Space Theory was written is precisely to give what you refer to as an 'honest' introduction to the foundations of analysis.

Edit: OP's comment below clarifies this book is not at the intended introductory level.

Ittay Weiss
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    -1: This is a very poor self publicity. This book has got nothing to do with introductory real analysis of relevance to the student. – Adhvaitha Nov 30 '14 at 20:46
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    can you please explain why you make that claim @Adhvaitha? – Ittay Weiss Nov 30 '14 at 20:46
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    The OP wants an introductory real analysis text, by which he needs some text that talks about

    1. Completeness of $\mathbb{R}$

    2. Sequences and series

    3. Continuity

    4. Derivatives

    5. Integral (Riemann)

    6. Some basic topology on $\mathbb{R}$

    Your book at best can be regarded as an "introductory" functional analysis book.

     – Adhvaitha Nov 30 '14 at 20:52
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    @Adhvaitha your negative comment appeared within a minute of the appearance of the answer, which means (as you obviously did not read the book) that you had time to just see the title. If you had looked at the TOC you would have seen that the first three chapters are, respectively, thorough introductions to linear spaces, topological spaces, and metric spaces. The style is very much what OP is looking for (i.e., very detailed proofs with little left for the reader). It covers completeness, topology, sequences, continuity. Rudin is given as benchmark. – Ittay Weiss Nov 30 '14 at 20:54
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    Sicne Rudin was mentioned I assumed OP is not looking for an introduction to elementary analysis of single valued real functions. I'm waiting for OP's clarification on this. – Ittay Weiss Nov 30 '14 at 20:57
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    My negative comment as you can see from the time stamp appeared $5$ minutes after your answer. An introductory real analysis text (by which I mean analysis on $\mathbb{R}$ instead of $\mathbb{R}^n$ or any other space) should first study the properties of $\mathbb{R}$ in detail, instead of topological spaces, linear spaces etc. It needs to first develop $\mathbb{R}$ and discuss all the topics such as convergence, continuity, derivative, etc. *in the context of $\mathbb{R}$*. A student reading real analysis for the first time should not be thrown away by premature abstraction. – Adhvaitha Nov 30 '14 at 21:00
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    +1 because it seems like a worthwhile reading to me. I need to clarify that my friend has just started to study real analysis and the course program is the same as the one which I followed: it starts from the very basics, goes to real valued functions of one variable, but has some references to topological concepts and metric spaces. – Dal Nov 30 '14 at 21:09
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    @Dal thank you for the clarification. Please indicate to me if you think I should delete the answer. – Ittay Weiss Nov 30 '14 at 21:45
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    @IttayWeiss Please, keep the answer here, because -- judging from the sample pages on linear spaces -- the book seems well-explained and therefore the parts on metric and topological spaces may prove useful for my friend (and to me too). – Dal Nov 30 '14 at 21:50
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    There are more a million topological spaces you could start with; there's no need to start every book with the real numbers. – Christopher King Dec 01 '14 at 04:09
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    @PyRulez The OP is asking for a "real" analysis text and not an analysis text starting with some topological space. – Adhvaitha Dec 02 '14 at 17:48
  • @Adhvaitha that doesn't make of Ittay's textbook bad, it is indeed of interest as someone will surely encounter metric spaces and topological spaces in real analysis, so one would choose an introductory real analysis text that deals with functions, sequences... then take his text which is the best treatment i found so far of metric spaces ... – user153330 Jan 31 '15 at 12:38
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    @Adhvaitha i find your negative comments more a way to dismiss self publicity than the content of the text itself, you didn't even look at sample pages – user153330 Jan 31 '15 at 12:39
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I have had the pleasure to teach introductory real analysis from couple of excellent texts, which I would also recommend.

Adhvaitha
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    +1: Although Bartle/Sherbert do make some "exercise to the reader" and "obvious" statements, it is a good text. – Clarinetist Dec 01 '14 at 01:26
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    Ross's book frequently appeals to the reader's naive understanding of calculus. Not sure if this is customary, but it might not be what OP has in mind in terms of "honest." Not sure, but definitely not a bad book - I used it in my first exposure. – A. Thomas Yerger Dec 01 '14 at 06:46
  • Bartle-Sherbert's very good for freshmen. In Real Analysis at my university all the professors I know endorse it as the main reference book. –  Sep 11 '20 at 17:21
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Bartle's Introduction to Real Analysis has a small number of "left to the reader" proofs, from I have seen so far.

5

Advanced Calculus by Patrick M. Fitzpatrick is a great text that starts from the very basics and goes up through point-set topology and metric spaces.

It starts with field axioms and builds from there so it doesn't "cheat" in that regard (edit: just to clarify, this means almost NO proofs are "left to the reader," the only exceptions being very small special cases that are then presented in exercises), but it doesn't contain solutions--I think you'd be hard pressed to find a textbook at that level that had complete solutions to every single problem. However, it's popular enough that most of the solutions can be found on this site or elsewhere on the internet.

Starting from general one-dimensional stuff and moving up to metric spaces in a single course seems like quite a task though.

ben
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Get two volumes of Zorich.$ $

Artem
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    I accept this answer because these are the books that my friend finded the most suited for the purpose. However, I would like to emphasize that I personally appreciate every suggestion given here and that some of the books mentioned will likely prove useful for me too as a complement to Rudin. Once again, thank you everyone! – Dal Dec 03 '14 at 21:53
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Terence Tao's Analysis I and Analysis II. These books are expanded and cleaned up versions of lecture notes which you can find here and here.

Henrik
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I used Rosenlicht's Introduction to Analysis in my real analysis course in undergrad. It's cheap and is a lot easier to digest than Rudin.

David Richerby
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Math1000
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For an introduction more 'foot on the ground' analysis I recommend Elementary Classical Analysis. It's a shame that the partial visualization is not available. But you will not regret in search for this book in the library of a good university.

See too Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach. By John Hubbard and Barbara Burke. More in Amazon site.

Elias Costa
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Yet Another Introduction to Analysis by Bryant is my favourite. It probably doesn't meet all the criteria you listed. However, it is the most intuitive first book on the topic I know, and once you have read it, other analysis books become much easier.

Mark Joshi
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