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Perhaps this is a repeat question -- let me know if it is -- but I am interested in knowing the best of Dover mathematics books. The reason is because Dover books are very cheap and most other books are not: For example, while something like Needham's Visual Complex Analysis is a wonderful book, most copies of it are over $100.

In particular, I am interested in the best of both undergraduate and graduate-level Dover books. As an example, I particularly loved the Dover books Calculus of Variations by Gelfand & Fomin and Differential Topology by Guillemin & Pollack.

Thanks. (P.S., I am sort of in an 'intuition-appreciation' kick in my mathematical studies (e.g., Needham))

EDIT: Thank you so far. I'd just like to mention that the books need not be Dover, just excellent and affordable at the same time.

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@Peter: Because he doesn't wear a scarf and a turtleneck to cover his chin. :-) –  Asaf Karagila Apr 12 '13 at 23:56
Also, for anyone interested in group theory, two great books by Passman have been recently added to the Dover catalog: amazon.com/Permutation-Groups-Dover-Books-Mathematics/dp/… and amazon.com/Algebraic-Structure-Group-Rings-Mathematics/dp/… –  user641 Apr 13 '13 at 18:25
As far as I'm aware, Guillemin and Pollack is only published by Prentice-Hall! –  AnonymousCoward Jul 8 '14 at 17:32

22 Answers 22

Though it lacks any treatment of cardinal functions, Stephen Willard’s General Topology remains one of the best treatments of point-set topology at the advanced undergraduate or beginning graduate level. Steen & Seebach, Counterexamples in Topology, is not a text, but it is a splendid reference; the title is self-explanatory.

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Pinter's A Book of Abstract Algebra is a great introductory text!

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Nathan Jacobson's Basic Algebra I is pretty good, along with the sequel for the more brave of heart. (Disclaimer I haven't read II, but I imagine it is also good).

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How does this book compare to Pinter's Book of Abstract Alegbra, also on Dover? –  frankster May 30 '14 at 13:10
@frankster I haven't read Pinter's book, but judging from the table of contents it seems to cover a bit less material in more detail than Jacobson. They look to be written at similar times but Pinter's does seem slightly more modern somehow. Hopefully someone who has read both can comment. –  Alex J Best May 30 '14 at 15:20

In my opinion the best nooks are:

"Ordinary Differential Equations" by Tenenbaum.

"Partial Differential Equations for Scientists and Engineers" BY Farlow

"Fourier Series and Orthogonal Functions" by Harry Davis

"Concepts of modern Mathematics" by Ian Stewart

There is a nice introduction to mathematics "Mathematics for the Nonmathematician" by Morris Kline. You will find also lots of books on maths for fun like "The Moscow Puzzles" which is great. And don't forget the classic "Flatland" -it's just a story of two dimensional creatures-.

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And I forgot to add: "Mathematics: Its Content, Methods and Meaning" by Kolmogorov. Tons of information here. –  Ambesh Apr 12 '13 at 19:38
I think this last book is 1000 pages or so. More about intuitive understanding that calculations. Interesting. –  Ambesh Apr 12 '13 at 20:08

I have picked up quite a few of those Dover editions over the years, and these are some currently on my bookshelf, together with my (probably very biased opinions):

Riesz & Nagy - Functional Analysis (dated but superbly well-written)

Katznelson - Introduction to Harmonic Analysis (also a bit dated, but very good)

Knopp - Theory and Application of Infinite Series (very dated, but still useful as a reference for anything to do with series)

Cohn - Advanced Number Theory (very useful as additional reading for anyone interested in the material covered by Cox's book on primes of the form $x^2+ny^2$)

Edwards - Riemann's Zeta Function (still worth reading for anyone interested in analytic number theory)

Pollard - Theory of Algebraic Numbers (might have been good in its day, but there are many better modern treatments)

Cassels - Rational Quadratic Forms (a very detailed treatment by a real master, but a tough read)

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Since it hasn't been mentioned yet: the Dover book that is most useful to me is none other than Abramowitz and Stegun's handbook. Though we now have the DLMF, there are still pieces of useful information within the Handbook, and even the DLMF still sometimes refers to it.

Otherwise, I have liked the numerical methods textbooks that have been republished by Dover: Ralston/Wilf and Hildebrand are among the nicer ones.

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I really like Goldblatt's book Topoi: The Categorial Analysis of Logic. I think it's a good introduction to category theory.

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Jech The Axiom of Choice is probably the book I have used the most in the last two years, and the best 15 USD I ever spent, not including beer (I even got my advisor and another professor from the department to order copies).

Now I ordered Moore's book about the axiom of choice, but that one is more of an historical overview. I didn't get the book yet, so I don't know how much I am going to use it, but it was definitely worth the 12 USD.

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Just out in Dover is Applied Analysis by the Hilbert Space Method, an undergraduate text on 2nd order linear differential operators by Prof. Sam Holland. I took his class at UMass Amherst when I was a sophomore in the late '80's/early '90's, and this text was then a bunch of typewritten notes. It is very inexpensive compared to the original Marcel Dekker version. I cannot recommend this text highly enough as a gateway to higher applied mathematics.

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I really like François Trèves's Topological vector spaces, distributions and kernels. I always give this book as reference when I answer some questions on TVS here or elsewhere.

Torgny Lindvall's Lectures on the coupling method is also a good book if you like probability theory.

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For a cheap non-Dover text, there is the new edition of Kunen's Set Theory. The first edition was published by North Holland/Elsevier and is almost prohibitively expensive (the softcover version still has a list price of over US\$100), except it was the standard text for graduate-level set-theory. The new edition is completely revised, slightly expanded, and sold at Dover-like prices.

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Oh, that’s nice. However, the original $1980$ North Holland/Elsevier edition is hardcover; I’ve a copy in arm’s reach. I think that the matching paperback came out three years later. It seems to go new for under $\$100$ despite the nominal list price of $\$104$. –  Brian M. Scott Apr 13 '13 at 6:35
@Brian: Even if amazon.com gives you a US\$26 discount on the first edition, it's still more than three times as expensive as the second! And for what? A slightly nicer font? (Hopefully I've corrected the mis-statements in my original answer.) –  Arthur Fischer Apr 13 '13 at 7:38
Oh, I wasn’t suggesting that it was a better buy than the new one; I was just curious to see what it was actually going for. As I said, I’m delighted to see it available at the price of a genre fiction hardcover novel! –  Brian M. Scott Apr 13 '13 at 7:41

Here's my personal best of, omitting things already mentioned:

  • The Undecidable, ed. Martin Davis -- collection of reprints of classic papers on computability and undecidability including Gödel, Turing, Church, Rosser, Kleene, Post
  • Nonstandard analysis, Robert. Intro to nsa using Nelson's internal set theory.
  • Regular polytopes, Coxeter. Lots in there that's hard to find elsewhere even after so much time.
  • Asymptotic methods in Analysis, de Bruijn
  • Asymptotic expsansions, Erdélyi (spelled wrongly on the spine)

There are Dover editions of Heath's Euclid and Archimedes, and lots of problem books (Mosteller's 50 challenging problems in probability, Dorrie's One Hundred Great Problems of Elementary Mathematics, Jacoby's Intriguing Mathematical Problems, Kratchik's Mathematical recreations,...).

Honourable mention also for Dubins and Savage How to gamble if you must, boringly retitled Inequalities for stochastic processes in its Dover edition.

I $\heartsuit$ Dover books.

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Chartrand Introductory Graph Theory

Suppes Axiomatic Set Theory

Kamke Theory of Sets - Nice introduction.

Smullyan Satan, Cantor and Infinity - Anything by Smullyan!

Rosenlicht Introduction to Analysis

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Number Theory, by George E. Andrews is a wonderful introduction to the subject for the beginner, especially the theory of partitions.

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I would add love for C.H. Edwards Advanced Calculus of Several Variables. I've used this as the text for an advanced calculus course a couple times. I also enjoyed using Lawden's Introduction to Tensor Calculus, Relativity and Cosmology which includes some intuitive calculations from physics. Although, it may need an update physically speaking in view of whatever is the current status quo with the cosmological constant.

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I used Edwards recently. It's really good, but it needs to be edited. It has weird inconsistencies in the theory and some straight-up (minor) errors. –  Ryan Reich Sep 29 '14 at 23:31
For example? (not meaning to argue, just curious about your take on it) –  James S. Cook Sep 30 '14 at 6:11
I left an Amazon review with some specifics. Basically, he is inconsistent with his $C^k$ hypotheses for Taylor's theorem and the mean value theorem. Sometimes his too-strong hypotheses are seemingly motivated by a desire to use an unnecessary technique in a proof. In the integral chapters, he has several proofs that are as horrible epsilon-wise as I've ever seen, again unnecessarily, since he could split them into several intuitive simple lemmas. –  Ryan Reich Sep 30 '14 at 23:53
Nice review. I'm curious, do you know of a book which has the improved proofs you mention in the review. I found his discussion of how the contraction mapping sets-up the inverse and implicit mapping theorems was pretty lucid. Maybe you've written something? –  James S. Cook Oct 1 '14 at 4:25
I wish I had either! But I just fixed things as I went along and didn't write formal notes, which I thought would take too long. My feeling was that the entire book needed to be redone, and not being an analyst, nor even having a fully formed opinion until the end of the class, I didn't think I was the one to do it. Perhaps next time I teach it. But, if you like, we could discuss the corrections that I remember making, by email. It probably wouldn't be too hard to reproduce them. –  Ryan Reich Oct 1 '14 at 5:25

coming from more of a computer science angle I find Fielder's "Special Matrices and Their Applications in Numerical Mathematics" really useful.

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After many unsuccessful attempts at self-learning, my real progress in tensor calculus developed from Lovelock's Tensors, Differential Forms,....-a Dover book. I really love that book.

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Rosenlicht's Intro to Analsysis was an awesome read, but the real learning took place in the excersises. It was cheap, and just as rigorous as the introductory analysis course I took the following semester!

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Elements of the Theory of Functions and Functional Analysis - Kolmogorv and Fomin This by far the best for me! (includes both the volumes!)

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