Books that every student “needs” to go through

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, not their yet I am on a four year course now had a year gap between the penultimate and final year)?

More focus on Algebra, Linear Algebra and Categories – Analysis, Set Theory, Measure theory (an area I have seen too little books dedicated for) also welcome.

I have searched these areas, though elementary text for self learning keep baring their heads, not the thing I’m looking for. I am looking for a text to ‘ put it all together ‘.

i.e. Spivak is very good for self learning basic real analysis, but Rudin really cuts to the heart.

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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:

  1. 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).
  2. 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.
  3. 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.
  4. 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:

  1. Lorenz and Levy–Read chapters 28-32, second only to Iwasawa, but with a different flavor (cohomological vs. formal group laws)
  2. Tate and Artin–The classic book. A little less readable then any of the alternatives here.
  3. 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.
  4. Iwasawa (read the rest of it!)
  5. 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

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.

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!

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.