Question

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.

Answer 1

The Chicago undergraduate mathematics bibliography is a nice annotated list of books.

Answer 2

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.

Answer 3

Spivak, Calculus

Answer 4

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

Answer 5

Kelley, General Topology

Answer 6

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.

Answer 7

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

Answer 8

Algebraic Topology by Hatcher (available online here).

Answer 9

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.

Answer 10

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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Answer 11

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

Answer 12

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.

Answer 13

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

Answer 14

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.

Answer 15

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.

Answer 16

Real Mathematical Analysis by Charles Pugh

Answer 17

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

Answer 18

Ordinary Differential Equations by Vladimir I. Arnold

Answer 19

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

Answer 20

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.

Answer 21

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.

Answer 22

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.

Answer 23

Basic Algebra by Jacobson.

Answer 24

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

Answer 25

Also, I just started this book and absolutely love it

Geometry: Euclid and Beyond, Hartshorne

Answer 26

Serre, A Course in Arithmetic.

Answer 27

Milnor, Topology from the differentiable viewpoint.

Answer 28

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

Answer 29

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".

Answer 30

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.