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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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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

For undergraduate level topology (mostly point set topology) I recommend "Topology" by Munkres. I learned topology from this book as an undergrad and I remember this being one of my favorite books at the time.

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Spivak, Calculus

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I agree, with the caveat that the subject of the book is usually called "Elementary Real Analysis" these days. That said, when I first read this book I loved it so much that it made me want to be a mathematician. It's both rigorous AND intuitive, in a way that both qualities complement one another. –  John Goodrick Oct 16 '09 at 21:46

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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Ok, this is not a single book, but I have often found books from the Springer Undergraduate Mathematics Series (SUMS) to be excellent. Here is a list of titles.

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Another one I like is "An introduction to Lie algebras." by Erdmann and Wildon.

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Algebraic Topology by Hatcher (available online here).

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If you like print books (easier to carry around, scribble in, etc.), Hatcher's book sells for $37, which seems pretty reasonable. –  Michael Lugo Oct 16 '09 at 20:12

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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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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Not exactly one of the topics in the question, but I particularly liked Silverman and Tate's Rational Points on Elliptic Curves.

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For discrete mathematics, I would recommend Van Lint-Wilson's "A Course in Combinatorics" as a good introductory text. It consists of 38 (in my edition) chapters that give (often largely self-contained) introductions to various areas of the field. Although it doesn't go nearly as in depth as, say, Stanley's "Enumerative Combinatorics" or a text focused solely on graph theory, I found it excellent for giving a broad overview and indicating to me where I wanted to explore deeper.

My one caveat would be that some chapters require background in either linear algebra or basic group theory, though those are easily skippable due to the structure of the book.

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The Princeton Lectures in Analysis by Stein and Shakarchi are great introductions to Fourier, complex, and real analysis (in that order!).

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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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Generatingfunctionology by Wilf is fun, free, requires very little in the way of prerequisites, and is as good an introduction to the methods of analytic combinatorics as could be asked for. It's long been one of my favorite textbooks.

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The free edition is the second (early 1990s?); there's a third edition (2005), which as of now is not free. The third edition doesn't differ that much from the second one, though. For a more advanced book that massively expands upon Wilf, I recommend Flajolet and Sedgewick's <i>Analytic Combinatorics</i>, published 2009 (also available <a href="algo.inria.fr/flajolet/Publications/AnaCombi/… online!</a>) -- but this is really a graduate-level text. –  Michael Lugo Oct 17 '09 at 5:11

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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Ordinary Differential Equations by Vladimir I. Arnold

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Concrete Mathematics, Graham, Knuth and Patashnik. Extremely useful, very good exercises, and a sense of humor that appeals to me.

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I finally gave in and bought this book last week, after realizing that at any given moment over the last few years I was more likely to have it checked out of the library than not. –  Michael Lugo Oct 17 '09 at 5:04

Introduction to Topology: Pure and Applied, by Adams and Franzosa. The figures in the book are beautiful, the problems are good, and the applications are good (and unusual) to see in an undergraduate text.

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Real Analysis, by Frank Morgan. The chapters are short and very directed. The proofs are written well. The exercises are well-selected. The book is written at a level accessible for most students.

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I am surprised no one has mentioned Halmos' Naive Set Theory or Finite-Dimensional Vector Spaces or Rudin's Principles of Mathematcal Analysis. There's also Sheldon Axler's Linear Algebra Done Right and Royden's Real Analysis.

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I believe we are supposed to put just one book per answer. It makes it easier to wote up. –  Grétar Amazeen Oct 17 '09 at 18:18
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Sheldon Axler (not Steven), for what it's worth. –  Tom Leinster Oct 24 '09 at 2:48

Basic Algebra by Jacobson.

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Volume I is a great guide for the advanced undergraduate, but I think that volume II is beyond what all but the most sophisticated undergraduates can deal with. –  David Speyer Oct 30 '09 at 13:05

Since Numerical Mathematics has not been covered, I would recommend the following

Introduction to Numerical Analysis by Stoer et. al. http://www.amazon.com/Introduction-Numerical-Analysis-J-Stoer/dp/038795452X/ref=sr_1_14?ie=UTF8&s=books&qid=1255807973&sr=8-14

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Also, I just started this book and absolutely love it

Geometry: Euclid and Beyond, Hartshorne

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Serre, A Course in Arithmetic.

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Milnor, Topology from the differentiable viewpoint.

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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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A basic undergrad algebra book which I feel is not as well known as it should be is Michael Artin's Algebra. I have it in soft cover so I hope it's actually the one in this Amazon link. Anyway it's beautifully written, provides context and motivation and is just a pleasure to read or browse. How often do you find a basic text written by a world-class expert?

"Always study from the masters".

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As far as I know there is only one book by Michael Artin with that title. I have the hardcover and it looks like the one you link to. Apparently Artin is working on a new edition. –  Michael Lugo Nov 1 '09 at 17:15
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That book is so much better than Dummit and Foote for undergraduates. D&F is also useless at the graduate level, where much better texts like Lang blow it out of the water. –  Harry Gindi Nov 30 '09 at 12:16
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Artin is going to be rough going for undergraduates who are not well versed in basic geometry and linear algebra,fpqc.But you can't help but love the infectious passion with which Artin weaves his craft in front of the students.He loves algebra and he's trying to prosyletize his students to it. A book with a similar geometric bent,level and also by a master that students will probably find easier going is E.B.Vinberg's A COURSE IN ALGEBRA. But Artin's book is very good and it's good news for all of us that Artin is revising it. –  Andrew L Mar 27 '10 at 22:01

As an undergraduate I loved Shafarevich's book Basic notions of algebra. This is not a textbook, but gives small beautiful tastes of a broad choice of topics in algebra, emphasizing connections with other fields.

I found it very stimulating, in the sense that every example or overview of some topic in this book made me want to learn more details about it. In fact I became interested in algebraic geometry because of this book.

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