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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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96 Answers 96

A Field Guide to Algebra by Antoine Chambert Loir. Covers a surprising amount of material considering the short length and minimal prerequisites

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"TOPOLOGY WITHOUT TEARS" by SIDNEY A. MORRIS is a very nice introduction in topological spaces theory, I think.

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Available online: uob-community.ballarat.edu.au/~smorris/topology.htm –  sdcvvc Nov 9 '09 at 14:09

Subject: FUNCTIONAL ANALYSIS

Erwin Kreyszig

Introductory Functional Analysis with Applications

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Geroch, Mathematical Physics

Don't be scared by the title: it teaches algebra, topology and measure theory, using category-theoretic language.

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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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"Differential Calculus on Normed Linear Banach Spaces" by Kalyan Mukherjee.

The author is a prof at the Indian Statistical Institute (Kolkata)

This is an amazing book which introduces differential calculus in arbitrary finite dimensional spaces (thinking of the derivative as the Jacobian) as the next leap from the Apostol and Rudin level and also build in topological ideas. It then goes into manifold theory and shows how to compute tangents to curves inside lie groups. It has nice sections of things like differentiating the determinant function and the matrix multiplication function and inverse function theorem and idea of equivalence of norms.

I would strongly recommend this book to an undergrad after he/she has done the Apostol/Rudin level of calculus.

"Calculus on Manifolds" by Spivak and "Differential geometry and Lie groups" by Kumaresan are two other good books which can be read alongside it.

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I think that Linear Algebra by Friedberg, Insel, and Spence is a spectacular linear algebra book. It gets straight to the point, it provides a worked out example or two exactly when they're needed, and it has lots of interesting exercises.

There are way too many gigantic linear algebra books with colour pictures and contrived examples which often seem to obsfucate the concepts being introduced. In my mind this book is 'the mathematician's linear algebra book'; clean and concise.

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Visual Complex Analysis by Tristan Needham is awesome!

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In fact, there are others that agree with you: the book is already in the list! –  Bjorn Poonen Mar 7 '10 at 7:46

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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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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I liked Elements of Abstract Algebra by Allan Clark, which is mainly a problem book with a moderate amount of exposition, but the problems are so well-chosen that a diligent undergraduate student working through all of them will come out with a solid knowledge of group theory, classical ring theory, and Galois theory.

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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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Linear Algebra: With Applications Otto Bretscher

Used at Carleton College – nice explanations, and quite a few proofs. Presents information primarily by providing examples, definitions/axioms and then proofs.

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I had many trials and these are in my opinion the best for an introductory, undergraduate level:

ODEs: Holzner: Differential Equations for Dummies
PDEs: Farlow: Partial Differential Equations for Scientists and Engineers

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G. J. O. Jameson: Topology and Normed Spaces for an introduction to functional analysis from a topological point of view.

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lindsey childs a concrete introduction to abstract algebra

here maybe.

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Strichartz, The Way of Analysis

Herstein, Abstract Algebra

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Milnor's book "Dynamics in One Complex Variable: Introductory Lectures". An early version is available from his website. Suitable for advanced undergraduates, graduate students, and mathematicians.

http://www.math.sunysb.edu/cgi-bin/preprint.pl?ims90-5

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Most readers here will not be able to appreciate them for a simple reason, but my favorite beginners' Analysis text is that by Bröcker. No-nonsense, concise, with a slight orientation towards topology.

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Hugo Steinhaus http://en.wikipedia.org/wiki/Hugo_Steinhaus book "Mathematical Kaleidoscope". It is kind of mathematical trivia sometimes very deep;-) It is not for learning math but for learning how to learn math in fun way.

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Complex Variables: Harmonic and Analytic Functions by Francis J. Flanigan

A nice little Dover paperback which turns the standard course on complex variables on its head. It begins by doing some multivariable calculus in the plane and harmonic functions, then proceeds to talk about complex numbers and to build analytic functions.

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I found «A (terse) introduction to linear algebra» (Katznelson) to be a much better book than Axler. It's part of the Student Mathematical Library and published by the AMS.

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For (applied) ODEs: Nonlinear dynamics and chaos by Steven Strogatz.

A very inspiring book! The explanations are crystal clear with lots of pictures. And it's funny too – the "Romeo and Juliet" illustration of 2-dimensional linear systems (Section 5.3) is a classic.

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Siegfried Bosch, Lineare Algebra

It's a very elegant, concise but beautifully written approach to Linear Algebra, and I love it.

Unfortunately for people who don't speak German, it has never been translated.

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I plan to post a complete reading list for undergraduates and graduate students at my blog this summer with my commentaries,but here's one I think that's available online and doesn't get nearly enough credit despite the fame of it's author: Gilbert Strang's Calculus. I wouldn't use anything else for a regular,non-honors calculus course. Carefully written,beautifully motivated with TONS of creative and SIGNIFICANT applications. I hope one day Strang finds the time to write a second edition-I have a list of improvements to suggest.

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A Concrete Approach to Abstract Algebra by W. W. Sawyer ($6 on Amazon!)

Though it goes a bit slow at times, it is by far the simplest, most intuitive book on Abstract Algebra in existence. Written for the non-mathematician, it does a great job of teaching the subject in simple, easy-to-understand prose. I couldn't put it down!

There are also two chapters on linear algebra, leading up to the final chapters, "vectors over fields" and "fields regarded as vector-spaces".

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I would recommend Walter Rudin's classic text entitled Real and Complex Analysis for the mathematically mature undergraduate student. (Of course, this should really only be read after one has familiarity with most of Rudin's earlier textbook: Principles of Mathematical Analysis.)

The beautiful thing about this book is that nearly every example or result stated by Rudin is used in "the big picture". As you progress through the book, you really start to appreciate the magic Rudin has used to weave together analysis in a manner that is rarely done in other textbooks.

While Rudin states in his preface that the textbook is intended as a first year graduate course on the subject of analysis, I believe that it is quite possible for a mathematically mature undergraduate to follow it. There is no assumption in the book that the reader has any familiarity with linear algebra, abstract algebra, general topology etc. beyond that which was covered in the first seven chapters of his earlier book, but realistically one would like to have at least mastered the basics of these areas before attempting to delve deeper into the analysis covered in this book. (There is a chapter on Banach algebras, but all the necessary abstract algebra here is developed from scratch.)

The textbook is indeed challenging with plenty of exercises, but it is not something with which a student with the correct prerequisites should have tremendous difficulty. Furthermore, if a student can successfully read most of the book, he is well-equipped to go deeper into most branches of modern analysis, and should find the other classic texts in the subject, for instance Royden's Real Analysis, Bartle's The Elements of Integration and Lebesgue Measure etc., very easy to follow.

The Amazon page for this book can be found here.

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Algebraic Theory of Numbers, by Pierre Samuel

Assumes only elementary knowledge of group and ring theory (even less than a complete undergraduate course which covers Galois theory) and develops algebraic number theory, a beautiful subject which puts much elementary number theory into an interesting perspective.

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