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Does anyone have a recommendation for a book to use for the self study of real analysis? Several years ago when I completed about half a semester of Real Analysis I, the instructor used "Introduction to Analysis" by Gaughan.
While it's a good book, I'm not sure it's suited for self study by itself. I know it's a rigorous subject, but I'd like to try and find something that "dumbs down" the material a bit, then between the two books I might be able to make some headway.
When I was learning introductory real analysis, the text that I found the most helpful was Stephen Abbott's Understanding Analysis. It's written both very cleanly and concisely, giving it the advantage of being extremely readable, all without missing the formalities of analysis that are the focus at this level. While it's not as thorough as Rudin's Principles of Analysis or Bartle's Elements of Real Analysis, it is a great text for a first or second pass at really understanding single, real variable analysis.
If you're looking for a book for self study, you'll probably fly through this one. At that point, attempting a more complete treatment in the Rudin book would definitely be approachable (and in any case, Rudin's is a great reference to have around).
I like Terrence Tao's Analysis Volume I and II. By his simple way of explaining things, this book must be readable by yourself.
You can see here http://terrytao.wordpress.com/books/ all his books along with the two, I mentioned above.
For self-study, I'm a big fan of Strichartz's book "The way of analysis". It's much less austere than most books, though some people think that it is a bit too discursive. I tend to recommend it to young people at our university who find Rudin's "Principle of mathematical analysis" (the gold standard for undergraduate analysis courses) too concise, and they all seem to like it a lot.
EDIT : Looking at your question again, you might need something more elementary. A good choice might be Spivak's book "Calculus", which despite its title really lies on the border between calculus and analysis.
Mathematical Analysis I & II by Vladimir A Zorich, Universitext - Springer. It has good number of examples and the explanations are lucid.
Bryant [1] would be my recommendation if you're fresh out of the calculus/ODE sequence and studying on your own. If your background is a little stronger, then Bressoud [2] might be better. Finally, you should take a look at Abbott [3] regardless, as I think it's the best written introductory real analysis book that has appeared in at least the past couple of decades.
[1] Victor Bryant, "Yet Another Introduction to Analysis", Cambridge University Press, 1990.
[2] David M. Bressoud, "A Radical Approach to Real Analysis", 2nd edition, Mathematical Association of America, 2006.
[3] Stephen Abbott, "Understanding Analysis", Springer-Verlag, 2001.
You might want to take a look at A Problem Text in Advanced Calculus by John Erdman. It's free, well-written and contains solutions to many of the exercises. These attributes, in my opinion, make it particularly well-suited for self-study. One of the things that I particularly like about the text is the author's use of o-O concepts to define differentiability. It simplifies some proofs dramatically (e.g., the Chain Rule) and is consistent across one-dimensional and n-dimensional spaces.
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"Principles of Mathematical Analysis" 3rd edition (1974) by Walter Rudin is often the first choice. This book is lovely and elegant, but if you haven't had a couple of Def-Thm-Proof structured courses before, reading Rudin's book may be difficult.
Thomas's calculus also seems to fit well to your needs, as i myself had used that book and found it more appealing than Rudin's
I've recently discovered Lara Alcock's 'How to think about analysis'. It isn't really a textbook, it's more of a study guide on how to go about learning analysis, but I believe it also covers the key ideas.
The book of Bartle is more systematic; much clear arguments in all theorems; nice examples-always to keep in studying analysis.
I recommend Mathematical Analysis by S. C. Malik, Savita Arora for studying real analysis. A very detailed and student friendly book!
I read this question a month ago and I decided to go for three of the most suggested books: Abbott "Understanding Analysis", Rudin "Principles of Mathematical Analysis", and Kolmogorov and Fomin "Introductory Real Analysis".
The one I liked most, and I ended up reading entirely, is Rudin's one: I am a PhD student in engineering and I think the level of the book was perfect to me. Two critiques I have are: there is a general lack of comments (a bit too much "Theorem, Proof") and there are no images. However, I found the book very clear and rigorous, especially the first 7 chapters. I definitely suggest it.
I really liked Abbott's approach: he really makes you understand the logic of things, and you never get lost in the proofs. On the other hand the one thing I didn't quite like was the excessive use of exercises: every two pages some kind of proof is "left to the reader." Sometimes also people that are not undergrads are going to read the book! Moreover this book treats only real numbers, and sometimes you lose the "big picture."
I stopped Kolmogorov and Fomin's book almost immediately. It was too much of an encyclopedia for me... But from the look I had, I bet it would be a great read if one has the time!
I really like Fundamental Ideas of Analysis by Reed. It's a friendly and clear introduction to analysis.
If you've had a strong course in Calculus, I highly recommend Advanced Calculus by G.B. Folland. It is well known that Folland's an amazing expositor; this book serves well to introduce you to the crucial transition from Calculus to Real analysis. This book should also prepare you sufficiently in terms of maturity for you to then be able to appreciate Baby Rudin.
1) Introduction to Real Analysis by mapa-
The contents are systematically structured with enough attention given to each topic. Some of the topics included in the book are Set Theory, Real numbers, Sets in R, Real Functions, Sequence, Series, Limits, Continuity and Differentiation. The book also contains solved exercises to help the readers understand the basic elements of the topics discussed in the contents
2) Elements of Real Analysis by denlinger
Two best books for self-study. Rudin and bartle are good if you have an instructor or in college but for self understanding these are best.
I recommend Courant and John's 'An introduction to Calculus and Analysis', volumes I and II. The authors give a rigorous treatment of their subject while still telling what motivates the ideas. Unlike many modern textbooks, they are not an sequence of definition-lemmas-theorems. These books emphasize ideas over structure. The authors' distinguished careers in applied mathematics ensures that there are plenty of examples and diagrams to illustrate their point.
Volume I focuses on calculus on the real line while volume II teaches functions of several variables. On their way, they teach exterior differential forms, ODE, PDE and elementary complex analysis.
Those with an 'applied' bent of mind, who want to trace the origin of ideas, not lose touch with the sciences that motivated development of mathematics may find these venerable volumes more rewarding than the modern treatments.
I think a good first book is 'A First Course in Mathematical Analysis' by David Alexandar Brannan and can suggest it as well as several that have already been mentioned on this page, but this one has the advantage that it was a byproduct of the Open University and is thus totally designed for self-study. Lots of problems placed near the relevant discussion, good margin notes for a beginner in analysis, and solutions to check your work.
If you still don't feel ready for Rudin after that, then I can recommend Alan Sultan's 'A Primer on Real Analysis' (which I'd recommend anyways because it should be better known) which is very nice and has lots of pictures to help development of intuition and lots of problems too with most solutions in the back of the book.
I'd also strongly recommend 'How to Prove It' by Daniel Velleman. You'll be writing proofs in Analysis and this is my favorite book in the proofs writing category. Very suitable to a beginner.
There are hordes of good books in all fields of mathematics. What you need is something you can learn from, not only the best and most glorious of these books.
Books with so many problems and exercises with their hints and solutions are very appealing. But what you really need is a mature and deep grasp of basics and concepts. After all, that's all you need to tackle these exercises with even a surprising level ease and fun.
Analysis is among the most reachable fields in math after high school, and a fair amount of knowledge is required in most of the other fields for beginners.
I do understand the emphasis on solutions because we all deal with self study, at least sometimes, and solutions/hints are crucial to make an evaluation of your own work. If you are really serious you will soon find out that what you really need are hints not solutions. Needless to say, hints or solutions are supposed to be a last resort, when there seems to be no way out. Even then, a hint is better taken only partially. And by the way: when tackling problems, it is when there seems to be NO WAY OUT that the actual LEARNING process takes place.
I encourage you to take a deep look into The Trillia Group funded, and free, Zakon's books: Mathematical Analysis I which followed by another volume, but to get some basics, Basic Concepts of Mathematics might be a good place to start. In the third mentioned book, this was mentioned:
Several years’ class testing led the author to these conclusions:
1 - The earlier such a course is given, the more time is gained in the follow- up courses, be it algebra, analysis or geometry. The longer students are taught “vague analysis”, the harder it becomes to get them used to rigorous proofs and formulations and the harder it is for them to get rid of the misconception that mathematics is just memorizing and manipulating some formulas.
2 - When teaching the course to freshmen, it is advisable to start with Sections 1–7 of Chapter 2, then pass to Chapter 3, leaving Chapter 1 and Sections 8–10 of Chapter 2 for the end. The students should be urged to preread the material to be taught next. (Freshmen must learn to read mathematics by rereading what initially seems “foggy” to them.) The teacher then may confine himself to a brief summary, and devote most of his time to solving as many problems (similar to those assigned) as possible. This is absolutely necessary.
3 - An early and constant use of logical quantifiers (even in the text) is extremely useful. Quantifiers are there to stay in mathematics.
4 - Motivations are necessary and good, provided they are brief and do not use terms that are not yet clear to students.
In the second book, this was mentioned:
Several years’ class testing led us to the following conclusions:
1 - Volume I can be (and was) taught even to sophomores, though they only gradually learn to read and state rigorous arguments. A sophomore often does not even know how to start a proof. The main stumbling block remains the ε, δ-procedure. As a remedy, we provide most exercises with explicit hints, sometimes with almost complete solutions, leaving only tiny “whys” to be answered.
2 - Motivations are good if they are brief and avoid terms not yet known. Diagrams are good if they are simple and appeal to intuition.
3 - Flexibility is a must. One must adapt the course to the level of the class. “Starred” sections are best deferred. (Continuity is not affected.)
4 -“Colloquial” language fails here. We try to keep the exposition rigorous and increasingly concise, but readable.
5 - It is advisable to make the students preread each topic and prepare questions in advance, to be answered in the context of the next lecture.
6 - Some topological ideas (such as compactness in terms of open coverings) are hard on the students. Trial and error led us to emphasize the sequential approach instead (Chapter 4, §6). “Coverings” are treated in Chapter 4, §7 (“starred”).
7 - To students unfamiliar with elements of set theory we recommend our Basic Concepts of Mathematics for supplementary reading. (At Windsor, this text was used for a preparatory first-year one-semester course.) The first two chapters and the first ten sections of Chapter 3 of the present text are actually summaries of the corresponding topics of the author’s Basic Concepts of Mathematics, to which we also relegate such topics as the construction of the real number system, etc.
I did not take these points very seriously, until I started reading and working on it. It is hard to find yourself completely stuck somewhere: it seems that all have been packed for a person who is learning on his own. Hints are provided whenever needed. In many occasions there are questions like "...Why?" which help in following the text rigorously.
I was recommended Introduction to Analysis by Mattuck. It was a bit difficult to use as it does not follow the progression other books (like Rudin or Apostol) follow. Maybe others can share more about their experience with this book, if they have used it.
Might not be a textbook but a very good supplement to a textbook would be the following book Yet Another Introduction to Analysis by Victor Bryant.
As a prerequisite the book assumes knowledge of basic calculus and no more.
This book may be a better starting point for some people.
For ones who read German, I strongly recommend Harro Heuser's 'Lehrbuch der Analysis Teil I'. There is also 'Teil II'. I tried couple of other German text books, but gave up continuing due to many errors or lack of completeness, etc. Then a person recommended me this book.
This book is self-contained and proofs are quite error-free as well as well-written for novices, though personally there were couple of proofs which were difficult to grasp, e.g. Cantor's Uncountability Proof and something else. The author tried to give proofs without the need of studying other subjects of mathematics, e.g. explaining compactness without referring to topology, which sometimes is a hard job. The author revised this book many times (lastest version is 17th edition). I feel sorry that the book has not been updated since the author has passed away in 2011. I recommend reading this book from the top to the bottom, even you have studied with another book before because the author builds up earlier proofs for later ones. I once tried to read from the middle, but gave up and re-started from the top.
The book also has good number of excercises and hints/solutions to selected problems at the end of the book, which I found good for self-learning.
This book assumes no prerequisites, but learnig other subjects parallely is always a good thing with math because it is hard to completely isolate a math subject from others.
I think Ross' Elementary Analysis: The Theory of Calculus is a good introductory text. It's very simple and well explained, but not quite at the level of Rudin's Principles of Mathematical Analysis (for example, everything is done using sequences in Ross, versus a general topological setting for open and closed sets in Rudin). But, if you master it, you can pick up the necessary ancillaries from Rudin or similar pretty quickly. FWIW, Rudin is the standard text for undergrad real analysis.
Another good option is Hoffman's Analysis in Euclidean Space. This was the book MIT used before Rudin arrived, and is a Dover book (so very cheap). I found its exposition to be comparable in level to Rudin, but easier to understand.
Finally, another book I can recommend is Hoffman's Elementary Classical Analysis. This is similar in level to Rudin, but has a lot more material worked out for you. Theres also a tiny bit on applications, so if you're an engineering/science student whose taking real analysis, it can be a bit helpful.
How "dumb" do you want it? I would say, at a university level at least, Steven R. Lay's book "Analysis - With an Introduction to Proof" is dumb vis-a-vis, say, a B student in an undergraduate honors analysis class:
Check the Amazon "first pages" preview to see the level it's at. Even if you don't get some of the stuff in the video I'm about to recommend I'd pair it with Harvey Mudd's YouTube series here, which you may already know about.
"Calculus" by David Patrick from "The Art of Problem Solving" book series is pretty good, and if your last exposure to the topic was in high school this book is actually much better than what's given in public high school and it comes from a problem solving standpoint, which I like because that is what math is used for, i.e., solving problems:
I would recommend "Understanding Analysis" by Stephen Abbott as well. I shall quote one paragraph that I like most.
In the first chapter, we established the Axiom of Completeness (AoC) to be the assertion that nonempty sets bounded above have least upper bounds. We then used this axiom as the crucial step in the proof of the Nested Interval Property (NIP). In this chapter, AoC was the central step in the Monotone Convergence Theorem (MCT), and NIP was the key to proving the Bolzano–Weierstrass Theorem (BW). Finally, we needed BW in our proof of the Cauchy Criterion (CC) for convergent sequences. The list of implications then looks like
AoC ⇒ NIP (&MCT)⇒ BW ⇒ CC.
But this one-directional list is not the whole story. Recall that in our original discussions about completeness, the fundamental problem was that the rational numbers contained “gaps.” The reason for moving from the rational numbers to the real numbers to do analysis is so that when we encounter a sequence that looks as if it is converging to some number—say √ 2—then we can be assured that there is indeed a number there that we can call the limit. The assertion that “nonempty sets bounded above have least upper bounds” is simply one way to mathematically articulate our insistence that there be no “holes” in our ordered field, but it is not the only way. Instead, we could have taken MCT to be our defining axiom and used it to prove NIP and the existence of least upper bounds. This is the content of Exercise 2.4.4. How about NIP? Could this property serve as a starting point for a proper axiomatic treatment of the real numbers? Almost. In Exercise 2.5.4 we showed that NIP implies AoC, but to prevent the argument from making implicit use of AoC we needed an extra assumption that is equivalent to the Archimedean Property (Theorem 1.4.2). This extra hypothesis is unavoidable. Whereas AoC andMCT canbothbeusedtoprove that N is not a bounded subset of R,there is no way to prove this same fact starting from NIP. The upshot is that NIP is a perfectly reasonable candidate to use as the fundamental axiom of the real numbers provided that we also include the Archimedean Property as a second unproven assumption. In fact, if we assume the Archimedean Property holds, then AoC, NIP, MCT, BW, and CC are equivalent in the sense that once we take any one of them to be true, it is possible to derive the other four. However, because we have an example of an ordered field that is not complete—namely, the set of rational numbers—we know it is impossible to prove any of them using only the field and order properties. Just how we decide which should be the axiom and which then become theorems depends largely on preference and context, and in the end is not especially significant. What is important is that we understand all of these results as belonging to the same family, each asserting the completeness of R in its own particular language. One loose end in this conversation is the curious and somewhat unpredictable relationship of the Archimedean Property to these other results. As we have mentioned, the Archimedean Property follows as a consequence of AoC as well as MCT, but not from NIP. Starting from BW, it is possible to prove MCT and thus also the Archimedean Property. On the other hand, the Cauchy Criterion is like NIP in that it cannot be used on its own to prove the Archimedean Property.1
I haven't started my first term yet, while I decide to self-study analysis. Initially I read Dexter Chua's lecture notes in "Numbers and Sets", then I read Terence Tao's analysis, but I am quite confused that they start from different initial definitions and starting points. "Understanding Analysis" perfectly solved my confusion and it illustrates concepts clearly.
I found Real analysis by Frank Morgan published by AMS a very nice introduction and Methods of Real analysis by Richard Goldberg a next one.
I would recommend "Guide to Analysis" by Hart & Towers which is aimed at those making the transition from high school mathematics to university mathematics and university analysis in particular. This seems like the most sensible choice.
However, the classic text to study real analysis would be "Principles of Mathematical Analysis" by Rudin. If you have not studied much mathematics before it may be tough going.
I enjoyed Introduction to Analysis by Maxwell Rosenlicht. I consider it a beautiful and elegant work. Some of the problems are rather difficult; but analysis is a difficult subject.
I had the pleasure of taking Differential Topology with him as an undergraduate at Berkeley. I thought he was pretty impressive. Also entertaining, with his "I'm getting all 'balled up'" comment from time to time.
I also liked Baby Rudin.
It's sad to see that nobody recommends the one I think is the best book ever written on introductory analysis: An Introduction to Classical Real Analysis by Karl Stromberg. I know... It's subjective.
Mathematical Analysis a straightforward approach by K.G Binmore is good for self study since it contains solutions to the exercises......The best book ever written is probably Introduction to Real Analysis by mapa.....