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What are some good undergraduate level books, particularly good introductions to (Real and Complex) Analysis, Linear Algebra, Algebra or Differential/Integral Equations (but books in any undergraduate level topic would also be much appreciated)?

EDIT: More topics (Affine, Euclidian, Hyperbolic, Descriptive & Diferential Geometry, Probability and Statistics, Numerical Mathematics, Distributions and Partial Equations, Topology, Algebraic Topology, Mathematical Logic etc)

Please post only one book per answer so that people can easily vote the books up/down and we get a nice sorted list. If possible post a link to the book itself (if it is freely available online) or to its amazon or google books page.

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  • $\begingroup$ This is borderline, but I think it is a legitimate question of interest to math instructors. I think its better as a community wiki though. $\endgroup$ – David Zureick-Brown Oct 16 '09 at 17:28
  • $\begingroup$ OK... Do I need to do something to make it a community wiki or is it already like that? $\endgroup$ – person Oct 16 '09 at 18:04
  • $\begingroup$ @person: Normally, to make a question or answer community wiki, you click the "community wiki" checkbox in the lower-right when you're composing or editing. But moderators have the power to convert a post to community wiki, which is what David has done here (actually, three moderators independently thought it should be converted to CW ... it's not something we do willy-nilly), so you don't need to do anything. $\endgroup$ – Anton Geraschenko Oct 16 '09 at 18:15
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    $\begingroup$ It's no longer possible to add useful answers to this question (as there are too many!) and it's unclear whether this question would be "allowed" by modern standards -- far too broad. As it's been popping back to the front page fairly frequently, we've decided to close it. $\endgroup$ – Scott Morrison Jul 11 '10 at 13:30
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    $\begingroup$ See discussion on meta: tea.mathoverflow.net/discussion/499/… (and remember to vote this comment up, so it is visible to others) $\endgroup$ – Victor Protsak Jul 14 '10 at 10:34

95 Answers 95

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I am surprised this has not been mentioned before (is it too advanced?):

Bott and Tu, Differential forms in algebraic topology.

The best introduction to de Rham cohomology, spectral sequences, characteristic classes from the algebraic point of view, and countless other topics.

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    $\begingroup$ Certainly true,but unless your undergraduates are in Germany or at Harvard,that book is definitely too tough for this list. $\endgroup$ – The Mathemagician May 25 '10 at 4:14
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    $\begingroup$ Hey, I was an undergraduate in Moscow. Does that count? $\endgroup$ – Victor Protsak May 25 '10 at 4:27
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    $\begingroup$ ... and seeing that Hatcher, Serre, Jacobson, Alperin, and Evans have been featured (some at the very top), I don't agree that it's "too tough for this list". $\endgroup$ – Victor Protsak May 25 '10 at 4:33
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    $\begingroup$ Yes-and with the POSSIBLE exception of Hatcher,none of them belong on a general reading list for undergraduates,Victor. You studied in a VERY strong program and you need to be a bit more mindful of that when making such lofty aspairations for mere mortals.And that goes for quite a few people in here. $\endgroup$ – The Mathemagician May 25 '10 at 11:55
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Galois Theory by Ian Stewart is excellent. The third edition is quite different from the second and includes many more problems.


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  • $\begingroup$ it has certain amount of typos (reflected by other students in bard). $\endgroup$ – Kerry Jul 22 '10 at 4:25
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Algebra: Chapter 0, Paolo Aluffi

Best book on algebra I've had my hands on yet, and I love how it uses category theory. I wouldn't mind having a course taught from this one. Topics from group theory all the way through field theory, linear algebra, and homology. This book deserves more attention!

https://www.amazon.com/Algebra-Chapter-Graduate-Studies-Mathematics/dp/0821847813/ref=sr_1_1?ie=UTF8&s=books&qid=1278799249&sr=8-1

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    $\begingroup$ Terrific book for first year graduate algebra or honors undergraduate. $\endgroup$ – The Mathemagician Jul 10 '10 at 22:23
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Jaenich: "Topology"

Introduces the concepts of point set topology ("paracompact" and all this stuff) motivating each via examples which are rigorously defined but also drawn. Other advantage: It is short!

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  • $\begingroup$ It's also great for someone trying to learn mathematical German! (not that that's terribly useful anymore... ><) $\endgroup$ – Michael Hoffman Oct 30 '09 at 13:31
  • $\begingroup$ I particularly love his proof of the Urysohn's lemma: the presentation of the main idea of the proof is brilliant, simple and clear. $\endgroup$ – Agustí Roig Oct 30 '09 at 19:31
  • $\begingroup$ I am SO totally pumped people are finally rediscovering this classic! $\endgroup$ – The Mathemagician Mar 18 '10 at 20:45
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Apostol "Calculus"

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Bartle "The Elements of Integration and Lebesgue Measure"

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I'm a big fan of John Hubbard's "Vector Calculus, Linear Algebra and Differential Forms" text. I was a TA for the course twice at Cornell and was amazed at how well it went. The text has an extremely pleasant "zest" to it. When Hubbard asked me to take a look at it my first response was the text is "overflowing with the spirit of calculus". I still believe that. I have a hard time containing my praise.

The main problem with the text is that it's so engrossing. It places more demands on the student than a traditional service course text would ever consider. But it's also far more rewarding. At Cornell it was taught as a branch of their traditional calculus sequence -- it was a course that was earmarked for keener students, mostly from other departments.

In short, if you want to have physics, engineering and economics students appreciating the derivative as a linear approximation, thinking Lipschitz bounds for functions are cool, being interested in the computation of norms of linear operators, etc, this is a great resource.

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  • $\begingroup$ There are "undergraduate" texts that are so deep with ideas and concepts that anyone of any level can learn from them. Books like this are Spivak's classic "Calculus On Manfolds",Janich's "Topology",Hoffman's "Analysis In Euclidean Space" and more recently McCleary's "A First Course In Topology:Continuity And Dimension". Hubbard and Hubbard CERTAINLY belongs in that select group. $\endgroup$ – The Mathemagician Mar 18 '10 at 20:31
  • $\begingroup$ John Hubbard is DA MAN!!! $\endgroup$ – Victor Protsak May 24 '10 at 6:05
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Searcóid: Elements of Abstract Analysis. I loved this book as an undergraduate, for many reasons, but mainly because it gave me an idea of the unity of mathematics. It starts from the axioms of set theory and takes you all the way to C*-algebras and the Gelfand-Naimark theorem. Here's the Google Books page.

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  • $\begingroup$ <pedant> The name is actually "Ó Searcóid" (cf. names like O'Grady, which are Anglicised versions of the same form).</pedant> Anyway, nice choice $\endgroup$ – user5117 May 25 '10 at 7:07
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Karen Smith et al., An Invitation to Algebraic Geometry

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    $\begingroup$ Since I haven't looked at this book but might be interested... This book is "undergraduate level" for whom? Presumably, many Harvard senior math majors would be able to tackle it. How many senior math majors at a mid-tier public research university? mid-tier liberal arts college? compass point state college? $\endgroup$ – Alexander Woo Dec 28 '09 at 21:23
  • $\begingroup$ (off-topic comment deleted) $\endgroup$ – S. Carnahan Aug 11 '10 at 13:45
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Real Mathematical Analysis by Charles Pugh

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    $\begingroup$ Thank you,someone finally mentioned this book.I'm hoping it supplants baby Rudin eventually.I affectionally call it "Rudin Done Right". $\endgroup$ – The Mathemagician Mar 18 '10 at 20:47
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Lectures on Linear Algebra by I. M. Gel'fand

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For a thorough introduction on Partial Differential Equations, read L.C. Evans, "Partial Differential Equations". Features both linear and nonlinear equations.

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  • $\begingroup$ that book is very good. For a 1st or 2nd year undergraduate, perhaps a slightly more accessible book is Haberman - "Applied Partial Differential equations"; admitted it is "applied", but it overlaps heavily with pure PDEs and has many "pure" techniques. $\endgroup$ – Vinoth Dec 21 '09 at 4:57
  • $\begingroup$ Evan's is a great GRADUATE level text. Most undergraduates would be like,"huh?" A much better choice is the long out of print Robert L.Street's "Analysis And Solution Of Partial Differential Equations". $\endgroup$ – The Mathemagician Mar 18 '10 at 20:34
  • $\begingroup$ @Andrew L: while the book is clearly meant for graduate students, it is also suitable for undergrads, especially the first chapters (and omitting the "omit on first reading" parts). With appropriate supervision, even some of the nonlinear chapters can be read by undergrads. $\endgroup$ – Martijn Mar 23 '10 at 7:37
  • $\begingroup$ @Martijn,those would have to be VERY good undergraduates indeed and you'd have to be REALLY selective with it. $\endgroup$ – The Mathemagician Mar 27 '10 at 21:48
  • $\begingroup$ That's quite a hefty book. I think undergrads would be better served by covering a less ambitious book rather than not getting very far into Evans. $\endgroup$ – John D. Cook Jul 10 '10 at 21:12
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From the quick look I've had, not much representation theory has been mentioned so here goes for undergrad level rep theory (perhaps suitable for 3rd/4th year in a standard sequence of undergraduate study), roughly in the order of difficulty (from easiest to hardest):

  • James & Liebeck - "Representations and Characters of Groups" (a very good introduction)

  • Sagan - "The symmetric group: representations, combinatorial algorithms, and symmetric functions"; (the first two chapters here at least are representation theory) OR James & Kerber - "Representation Theory of the Symmetric Group" (this one includes some modular representations of $S_n$)

  • Alperin - "Local Representation Theory" (basically, modular representation theory)

  • Hall - "Lie groups, Lie Algebras and Representation Theory" (a solid introduction to Lie theory); for a more advanced perspective Harris & Fulton - "Representation Theory: A first course" (but it could be slightly terse at points, but not necessarily)

For algebraic geometry, the one book I'd suggest is "Algebraic Geometry: A first course" by Joe Harris, very nice and full of examples. For algebraic number theory, a very good introduction is Janusz - "Algebraic Number Fields" (followed perhaps by Childress - "Class Field Theory", or Silverman - "The Arithmetic of Elliptic Curves" to go in a slightly different direction).

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Here is an undergraduate level math book recommendation from an early undergrad's position:

I like "Linear Algebra Done Right". I've looked at a bunch of books on linear algebra, and the usual matrix approach is to me a big turn-off when what you're really interested in is the abstract machinery of transformations between vector spaces. I'm not a research mathematician. In fact, I don't even study linear algebra yet, but as a student of mathematics that like algebra, spaces, maps and all that good stuff, I find this to be a very readable account of linear algebra.

There are more abstract books on the subject, and my impression is that LADR prepares you for the next level way before you're usually "allowed to" by other accounts like Lax etc. The trade-off is that LADR is not a book for engineers, but this would be a sad world for a mathematician if that was something he had to worry about (in his spare time). Great for self-study. Reads like a novel. I'd probably prefer it if Axler used sets for span and bases instead of lists, but that's something you'll probably be able to shake off with the next book you read on the subject.

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    $\begingroup$ See the other answer on this book for my comments. $\endgroup$ – Gerald Edgar Dec 28 '09 at 15:24
  • $\begingroup$ I'm not sure I agree your comments warrants a vote-down for this one. I still think it's a good book (for people who wants to learn mathematics for the benefit of mathematics), and my position for what I want the book to do is clear and I think it accomplishes it. $\endgroup$ – Eivind Dahl Dec 28 '09 at 15:43
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Visual Complex Analysis by Tristan Needham is awesome!

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    $\begingroup$ In fact, there are others that agree with you: the book is already in the list! $\endgroup$ – Bjorn Poonen Mar 7 '10 at 7:46
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Also, I just started this book and absolutely love it

Geometry: Euclid and Beyond, Hartshorne

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Fraleigh's "A First Course in Abstract Algebra"

https://www.amazon.com/First-Course-Abstract-Algebra-7th/dp/0201763907

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For a long time, Kolmogorov-Fomin's Introductory Real Analysis was my standard for a great mahtematics textbook. I can't imagine a better introduction to serious analysis.

The translation I'm linking to is very good, and includes excercises (the original has many fewer), but it is incomplete (it's missing the chapter on Fourier Series). So if you can read Russian, I recommend you get the original.

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  • $\begingroup$ You know,it's interesting you should approve so strongly of the Silverman translation of the Kolomogrov/Fomin text,Ilya. A lot of Russian mathematicians I've brought it up to tell me Silverman should be hanged for ruining such a classic.Guess you can't please everyone. $\endgroup$ – The Mathemagician Mar 28 '10 at 5:26
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Linear Algebra / Hoffman & Kunze - A book that truly develops linear algebra in a gradual manner. It starts with a basic discussion of systems of linear equations, matrices, Gaussian elimination, etc. and gradually progresses to the more abstract theory. Eventually it even touches upon subjects such as tensor products, the exterior algebra and the Grassmann ring. In short, it manages to cover a lot of linear algebra in a very leisurely and clear manner. I think that this is the quintessential example of a how an undergraduate level math book should be written. The only thing I don't like about it is the fact that quotient spaces aren't mentioned throughout the book (they're mentioned in the appendix, though).

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  • $\begingroup$ I'd LOVE to teach a one year honors course in linear algebra using the union of both H&K and Strang. A truly balanced course that shows how both aspects of the subject-theory and applications-are equally important. $\endgroup$ – The Mathemagician Jul 11 '10 at 0:30
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Subject: FUNCTIONAL ANALYSIS

Erwin Kreyszig

Introductory Functional Analysis with Applications

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E. Hairer, G. Wanner: Analysis by its history for an introduction to real and numerical analysis from a historical point of view.

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Kock, Vainsencher: An invitation to Quantum Cohomology.

Written in the most friendly and motivating style I have ever seen in a book. Almost has no prerequisites: You should that there exists something like algebraic varieties - without having to know any technical details - and that P^1 is such a thing. Everything else is provided in easy exercises or the text. It gives an excellent intuition about the subject with lots of outlooks on a field of current research, and at the same time manages to be easily undergraduate readable.

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Lebesgue Integration on Euclidean Space / Frank Jones - an extremely readable book on Lebesgue theory on $\mathbb{R}^n$ (lots of figures and geometric intuition). He constructs Lebesgue measure in a very down-to-earth manner, much more explicitly than other more abstract constructions (via Caratheodory's extension theorem or Riesz's representation theorem). In my experience, it's best to first study Lebesgue measure on $\mathbb{R}^n$ and only then point out that it's merely one instance of the general theory of measures, which is the way this book is written. It can't compare with the "tougher" books on measure theory (e.g. Big Rudin) since it doesn't discuss the Radon-Nikodym theorem and many other important theorems in measure theory, but then again the book is clearly intended for an undergraduate audience, and as for Lebesgue theory on Euclidean spaces, it provides a pretty complete picture.

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  • $\begingroup$ One of the best introductions to the subject out there. Should be on everyone's must-read list and great collateral reading with the more intense introductions like Big Rudin or Folland. $\endgroup$ – The Mathemagician Jul 11 '10 at 0:29
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Alexandre Stefanov keeps an extensive list (Wayback Machine, another link) of free math books / lecture notes. The list is divided according to subject and updated frequently. I have found some very nice books there.

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  • $\begingroup$ that link doesn't work! could you try posting it again? $\endgroup$ – Aaron Mazel-Gee Oct 23 '09 at 18:09
  • $\begingroup$ Thanks for letting me know of the broken link. It now works. $\endgroup$ – Henning Arnór Úlfarsson Nov 6 '09 at 21:48
  • $\begingroup$ Link which is currently working: trillia.com/online-math $\endgroup$ – Matthew Daws Jun 7 '18 at 21:53
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Kelley, General Topology

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Linear Algebra and Its Applications by Gilbert Strang. You can also watch his video lectures at MIT OpenCourseWare

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"Introduction to Mathematical Logic" by Ebbinghaus, Flum and Thomas

Careful introduction, addresses many doubts that one might have about why one does logic in this way and not some other, e.g. whether one is doing something circular when formulating set theory in 1st order logic, or e.g. it proves Lindstroem's Theorem, that says that classical 1st order logic has the highest power of expressability among the logics with completeness and Loewenheim-Skolem.

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How about an anti-recommendation? Someone in another answer mentioned Steven Axler's Linear Algebra Done Right. My comment, not as someone who has used this book in a class, but as someone who has taught the students from this class during the following term: It doesn't prepare the students to use linear algebra in engineering, in physics, in chemistry, or even in branches of mathematics other than abstract algebra.

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  • $\begingroup$ I suppose the correct thing to ask then is: what WOULD you use? Not to be confrontational, just that the standard around these parts is pretty poor, and Axler is considered much better. Also, could you explain what they're not teaching? I'd be interested to know! Thanks! $\endgroup$ – Michael Hoffman Oct 30 '09 at 13:27
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    $\begingroup$ Then how about "Linear Algebra Done Wrong", available for free at math.brown.edu/~treil/papers/LADW/LADW.html? $\endgroup$ – Henning Arnór Úlfarsson Nov 6 '09 at 21:47
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    $\begingroup$ @Gerald: I'm not sure omission of Cramer's Rule is such a big deal. Cramer's Rule is helpful for solving small linear systems (2 or 3 unknowns) since there are useful heuristics for calculating the determinants of 2x2 and 3x3 matrices, but Gaussian elimination is a more powerful and general algorithm for solving linear systems. $\endgroup$ – las3rjock Nov 7 '09 at 15:49
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    $\begingroup$ * Then how about "Linear Algebra Done Wrong", * Looks good, actually. Suitable for students interested in other branches of mathematics! $\endgroup$ – Gerald Edgar Dec 30 '09 at 13:35
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    $\begingroup$ The book is for students already familiar with matrix algebra, so this is not a valid criticism of the book. $\endgroup$ – Michael Greinecker May 25 '10 at 20:51

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