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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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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. –  Scott Morrison Jul 11 '10 at 13:30
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See discussion on meta: tea.mathoverflow.net/discussion/499/… (and remember to vote this comment up, so it is visible to others) –  Victor Protsak Jul 14 '10 at 10:34

96 Answers 96

Topics in Algebra by I. N. Herstein. A new edition will be coming out this year.

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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!

http://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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Terrific book for first year graduate algebra or honors undergraduate. –  Andrew L Jul 10 '10 at 22:23

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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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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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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... 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". –  Victor Protsak May 25 '10 at 4:33

Also, I just started this book and absolutely love it

Geometry: Euclid and Beyond, Hartshorne

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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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Real Mathematical Analysis by Charles Pugh

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Thank you,someone finally mentioned this book.I'm hoping it supplants baby Rudin eventually.I affectionally call it "Rudin Done Right". –  Andrew L Mar 18 '10 at 20:47

Lectures on Linear Algebra by I. M. Gel'fand

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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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Alexandre Stefanov keeps an extensive list 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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Apostol "Calculus"

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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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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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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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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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Karen Smith et al., An Invitation to Algebraic Geometry

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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? –  Alexander Woo Dec 28 '09 at 21:23

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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See the other answer on this book for my comments. –  Gerald Edgar Dec 28 '09 at 15:24

There is a good list here, divided by subject, that also contains many links to freely available textbooks and lecture notes.

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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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Metric Spaces by Mícheál Ó Searcóid

http://www.amazon.com/Metric-Spaces-Springer-Undergraduate-Mathematics/dp/1846283698/ref=sr_1_1?ie=UTF8&s=books&qid=1256082496&sr=1-1

It's an exhaustive introduction analysis at the level of the metric space that's well worth reading.

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Introduction to Analysis by William R. Wade

http://www.amazon.com/Introduction-Analysis-4th-William-Wade/dp/0132296381/ref=sr_1_1?ie=UTF8&s=books&qid=1256082707&sr=1-1

This is a good transition from undergraduate calculus to analysis.

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Godement "Analysis" (I,II,III,IV) http://www.amazon.com/Analysis-Convergence-Elementary-functions-Universitext/dp/3540059237

"... The content is quite classical ... [...] The treatment is less classical: precise although unpedantic (rather far from the definition-theorem-corollary-style), it contains many interesting commentaries of epistemological, pedagogical, historical and even political nature. [...] The author gives frequent interesting hints on recent developments of mathematics connected to the concepts which are introduced. The Introduction also contains comments that are very unusual in a book on mathematical analysis, going from pedagogy to critique of the French scientific-military-industrial complex, but the sequence of ideas is introduced in such a way that readers are less surprised than they might be.

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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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Then how about "Linear Algebra Done Wrong", available for free at math.brown.edu/~treil/papers/LADW/LADW.html? –  Henning Arnór Úlfarsson Nov 6 '09 at 21:47
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@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. –  las3rjock Nov 7 '09 at 15:49
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* Then how about "Linear Algebra Done Wrong", * Looks good, actually. Suitable for students interested in other branches of mathematics! –  Gerald Edgar Dec 30 '09 at 13:35
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The book is for students already familiar with matrix algebra, so this is not a valid criticism of the book. –  Michael Greinecker May 25 '10 at 20:51

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