What books must every math undergraduate read? I'm still a student, but the same books keep getting named by my tutors (Rudin, Royden).
I've read Baby Rudin and begun Royden though I'm unsure if there are other books that I "should" be working on if I want to study beyond Masters. I'm not there yet as I'm on a four year course and had a gap year between Years 3 and 4.
Please recommend for Algebra, Linear Algebra and Categories - Analysis, Set Theory, Measure theory (an area I have seen too little books dedicated for).
E.g. Spivak is very good for self learning basic real analysis, but Rudin really cuts to the heart.
 A: Alex Youcis's and other users suggestions are pretty good but I think some of the the books that they mentioned are a little difficult for average undergraduate students. But I totally agree with most of their suggestions.
Here are some books which I found really nice and helpful for undergraduate level:
Algebra :

*

*Algebra in Action: A Course in Groups, Rings, and Fields by Shahriar Shahriari

*Abstract Algebra: An Introduction with Applications by Derek Robinson

Linear Algebra :

*

*Linear Algebra Done Right by Sheldon Axler

Analysis :

*

*Real Mathematical Analysis by Charles C. Pugh

*Measure, Integration and Real analysis by Sheldon Axler (More Advanced)

Complex Analysis :

*

*Complex Analysis by Theodore W. Gamelin

*A Course in Complex Analysis and Riemann Surfaces by Wilhelm Schlag (More Advanced)

Algebraic Topology :

*

*Basic Topology by M. A. Armstrong

*Introduction to Topological Manifolds by John M. Lee

Algebraic Geometry :

*

*An Invitation to Algebraic Geometry by Karen E. Smith & others

Smooth Manifolds :

*

*An Introduction to Manifolds by Loring W. Tu

Elementary Number Theory :

*

*Elementary Number Theory and Its Applications by Kenneth H. Rosen

Elliptic Curves :

*

*Rational Points on Elliptic Curves by Joseph H. Silverman & John T. Tate

Dynamical Systems :

*

*Differential Equations, Dynamical Systems, and an Introduction to Chaos by Morris W. Hirsch, Stephen Smale & Robert L. Devaney

*Complex Dynamics by Lennart Carleson & Theodor W. Gamelin

*Dynamics in One Complex Variable by John W. Milnor

A: Here are a few of the books I've found especially rewarding:

*

*Linear Algebra - Friedberg, Insel, Spence
An excellent introduction to finite dimensional linear algebra. In fact, for most undergraduate stuff you won't need anything else.


*Principles of Mathematical Analysis - Rudin
Perfect intermediate textbook between typical undergraduate books and typical graduate books. The exposition to "Rudin-style books" is almost as valuable as the actual content.


*Mathematical Analysis - Apostol
Covers a few more topics than Rudin and is a little more explicit.


*Complex Analysis - Stein, Shakarchi
A little difficult as a first introduction but very good.


*Partial Differential Equations - Rauch
This book is geared at graduate students, but accessible to undergraduates with a strong background in my opinion. Its a great first introduction to "serious PDE" (i.e. not your typical cookbook course). Another good one is Introduction to Partial Differential Equations - Renardy & Rogers. This second one also does not assume any familiarity with Lebesgue integration, so it might be better for undergraduates.


*Introduction to the Theory of Groups - Rotman
This is a great second text in group theory after an elementary exposition (say Contemporary Abstract Algebra - Gallian).


*Real and Complex Analysis - Rudin
For those wishing to go further in analysis this is absolutely necessary as far as I'm concerned.
A: EDIT: I now think that this list is long enough that I shall be maintaining it over time--updating it whenever I use a new book/learn a new subject. While every suggestion below should be taken with a grain of salt--I will say that I spend a huge amount of time sifting through books to find the ones that conform best to my (and hopefully your!) learning style.

Here is my two cents (for whatever that's worth). I tried to include all the topics I could imagine you could want to know at this point. I hope I picked the right level of difficult. Feel absolutely free to ask my specific opinion about any book.
Basic Analysis: Rudin--Apostol
Measure Theory: Royden (only if you get the newest fourth edition)--Folland
General Algebra: D&F--Rotman--Lang--Grillet
Finite Group Theory: Isaacs-- Kurzweil
General Group Theory: Robinson--Rotman
Ring Theory: T.Y. Lam-- times two
Commutative Algebra: Eisenbud--A&M--Reid
Homological Algebra: Weibel--Rotman--Vermani
Category Theory: Mac Lane--Adamek et. al--Berrick et. al--Awodey--Mitchell
Linear Algebra: Roman--Hoffman and Kunze--Golan
Field Theory: Morandi--Roman
Complex Analysis: Ahlfors--Cartan--Freitag
Riemann Surfaces: Varolin(great first read, can be a little sloppy though)--Freitag(overall great book for a second course in complex analysis!)--Forster(a little more old school, and with a slightly more algebraic bend then a differential geometric one)--Donaldson
SCV: Gunning et. al--Ebeling
Point-set Topology: Munkres--Steen et. al--Kelley
Differential Topology: Pollack et. al--Milnor--Lee
Algebraic Topology: Bredon--May-- Bott and Tu (great, great book)--Rotman--Massey--Tom Dieck
Differential Geometry: Do Carmo--Spivak--Jost--Lee
Representation Theory of Finite Groups: Serre--Steinberg--Liebeck--Isaacs
General Representation Theory: Fulton and Harris--Humphreys--Hall
Representation Theory of Compact Groups: Tom Dieck et. al--Sepanski
(Linear) Algebraic Groups: Springer--Humphreys
"Elementary" Number Theory: Niven et. al--Ireland et. al
Algebraic Number Theory: Ash--Lorenzini--Neukirch--Marcus--Washington
Fourier Analysis--Katznelson
Modular Forms: Diamond and Shurman--Stein
Local Fields:


*

*Lorenz and Levy--Read chapters 23,24,25. This is by far my favorite quick reference, as well as "learning text" for the basics of local fields one needs to break into other topics (e.g. class field theory).

*Serre--This is the classic book. It is definitely low on the readability side, especially notationally. It also has a tendency to consider things in more generality than is needed at a first go. This isn't bad, but is not good if you're trying to "brush up" or quickly learn local fields for another subject.

*Fesenko et. al--A balance between 1. and 2. Definitely more readable than 2., but more comprehensive than 1. If you are wondering whether or not so-and-so needs Henselian, this is the place I'd check.

*Iwasawa--A great place to learn the bare-bones of what one might need to learn class field theory. I am referencing, in particular, the first three chapters. If you are dead-set on JUST learning what you need to, this is a pretty good reference, but if you're likely to wonder about why so-and-so theorem is true, or get a broader understanding of the basics of local fields, I recommend 1. 


Class Field Theory: 


*

*Lorenz and Levy--Read chapters 28-32, second only to Iwasawa, but with a different flavor (cohomological vs. formal group laws)

*Tate and Artin--The classic book. A little less readable then any of the alternatives here.

*Childress--Focused mostly on the global theory opposed to the local. Actually deduces local at the end as a result of global. Thus, very old school.

*Iwasawa (read the rest of it!)

*Milne--Where I first started learning it. Very good, but definitely roughly hewn. A lot of details are left out, and he sometimes forgets to tell you where you are going.


Metric Groups: Markley
Algebraic Geometry: Reid--Shafarevich--Hartshorne--Griffiths and Harris--Mumford
A: This may not be relevant to you, but for others who are still in high school or first and second year university the following book by Chartrand, Polimeni, and Zhang, is an incredible introduction to proofs and various areas of mathematics
"Mathematical Proofs: A Transition to Advanced Mathematics" by Gary Chartrand, Albert D. Polimeni, and Ping Zhang.
There is an entire chapter devoted to each of the following:


*

*Communicating Mathematics

*Naive Set Theory

*Logic

*Direct Proof

*Proof by Contrapositive

*Existence and Proof by Contradiction

*Mathematical Induction (and Strong Induction)

*Equivalence Relations (Equivalence Classes, Congruence Modulo n, Modular arithmetic)

*Functions (Bijective, Inverse, Permutations)

*Set Theory (up to Schroder-Bernstein Theorem and the Continuum Hypothesis)

*Number Theory

*Calculus (Limits, Infinite Series, Continuity, Differentiability)

*Group Theory (up to Isomorphic Groups)


With Three Additional Chapters online covering:


*

*Ring Theory

*Linear Algebra

*Topology


All with in depth worked out solutions to each chapter. I think every serious first-year mathematics student should work through this entire book thoroughly doing as many questions as possible!
A: Real Analysis - absolutely loved Wade's and Counterexamples in Analysis (with an analogue Counterexamples in topology). The first is good for an intro and the second is good for pedagogical purposes. Also I went through A Radical Approach to Real Analysis front to back doing every single problem in it, a very useful book.
Measure Theory - A Radical Approach to Lebesgue's Theory of Integration
Nice intro to (Galois) Group Theory (not a textbook by any means but still very well-written)
And in general, if you have a bit more time, throughout my last ten years of college life I have Schaum's outlines to be handy frequently. They have a book for topology and linear algebra and so on. The first few chapters can be a good intro. You might even get through the whole book on your own before actually taking the course (will take a long long time though). But during your course (as a supplement) or later on as a quick thorough review or reference, these books are very useful. I refer to them to this day for a quick property/theorem or a cool example/counterexample/solution when I can't find it on wikipedia.
True story, one of the questions on my PDE comprehensive exam for my PhD in math was lifted right out of Schaum's Outline for PDEs.
A: I second the endorsement of Schaum's Outlines.  
I actually differ from the view that you have to read iconic books to "hang".  If you read any book and do a good job on it (mastery learning) that will drive your comprehension and skills.  Perfection of Schaum's will trump mediocre work through a tougher text any day of the week.  In addition, there are always different tricks or approaches to find if you look at different books.
