Jacobson was my first real algebra book, and I retain an affection for it. The book is very densely written, and his prose has its own beauty but is difficult to get much from at first. The selection of topics is interesting: chapters 1–4 cover groups, rings, modules, fields (modules in the linear-algebra sense, that is, over principal ideal domains), while chapters 5–8 cover extension topics not usually found in general texts. He deliberately avoids modernist abstraction, preferring an explicit construction to a universal property and a commutative diagram (although the universal property is frequently given), and this complicates his notation and prose at times, especially in the module chapter. The field-theory chapter is fantastic. Some of the exercises are deliberately too hard.

Many people like this book, but I don't. Hungerford covers the standard topics from group, ring, module, and field theory, with a little additional commutative ring theory and the Wedderburn theory of algebras. The field-theory chapter is horrible, and the rest of the book is okay but doesn't excite me. (And the typesetting is bad.)

Well, do you like Serge Lang books, or not? Like every other Serge Lang book, this one is uncompromisingly modern, wonderfully comprehensive, and unpleasantly dry and tedious to read. Unlike most other Serge Lang books, this one has exercises, at least.

I keep recommending this book to people because it's the only hard one whose contents correspond well to the 257-8-9 syllabus, and also because I like Mac Lane's treatment of linear and multilinear algebra. Mac Lane and Lang are the only books in this group which treat multilinear (tensor) algebra at all, and believe me, you'll need it eventually. Worth a look to see whether you find Mac Lane's style congenial. Not to be confused with Birkhoff/Mac Lane, A survey of modern algebra (a much shorter and easier book).

[BR] I used Mac Lane/Birkhoff's book pretty heavily in Math 257 and 258. Unlike most algebra books I've seen, they don't put all the group theory at the beginning and all of the field theory at the end, but prefer to develop each topic a little bit at a time and then develop it with more depth later. As a result, this book is hard to use as a reference. You can't get past rings without tackling categories and universal constructions which are used heavily throughout the remainder of the text. However, their treatment of categorical algebra is one of the more readable introductions to the theory I've come across.

This is a linear algebra book written by a functional analyst, and the crux of the book is a treatment of the spectral theorem for self-adjoint operators in the finite-dimensional case. It's a beautiful, wonderful book, but not a very good reference for traditional linear algebra topics or applications. You also have to read a fair distance before you even see a linear map, and the exercises are mostly too easy, with a few too hard. But this book was where I first learned about tensor products, and why the matrix elements go the way they do and not the other way (Halmos is very careful on this point).

[PC] I own this book and read through it often, but it's never taught me linear algebra per se. Let's agree that it's too abstract for a reasonable first introduction to linear algebra; it's really meant for students who already know (some) linear algebra to read through and appreciate one particular, and particularly elegant, presentation of the material. If you want to know about the linear algebra which surrounds functional analysis, then by all means read this book, but much of the material is nonstandard and a bit curious from the perspective of mainstream linear algebra; projections seem to be the most important linear map, and there are many sections lovingly devoted to commuting projections, decomposing projections, etc. I still am not sure why Halmos deifies the [,] as much as he does, and quite honestly, I would learn multilinear algebra anywhere but here.

If you can stand terrible typesetting and an unexciting prose style, this tiny little book is a good rigorous reference for traditional linear algebra (i.e., it doesn't assume you're a tree). A nice bonus at the end is the Wedderburn theorem for division algebras over R, although the lack of sophistication makes for some unmotivated technical carpentry. I look in here whenever I can't remember what a positive-definite matrix is.

You may never need The Book on linear algebra. But one day, you may just have to know fifteen different ways to decompose a linear map into parts with different nice properties. On that day, your choices are Greub and Bourbaki. Greub is easier to carry. End of story.

The first half is a coherent, systematic development of elementary number theory, assuming the basics of algebra. In the second half the authors explore more advanced topics of an algebraic/geometric flavor (zeta functions, L-functions, algebraic number fields, elliptic curves). Lots of exercises. This book helped make number theory make sense to me. You will find many introductory number theory texts pitched below I/R, but if you can read I/R, ignore the easy ones.

[PC] Yes, this is the standard and to my knowledge the best number theory text that is modern, broad, and reasonably elementary. It's a strange book in that it's really not written at any one level—if you've heard of something called unique factorization, you'll find the first few chapters easygoing material, but the algebraic sophistication rises slowly but surely throughout the book. Eventually you need to be comfortable with rings, fields and Galois theory at the undergraduate level, but they tell you at the beginning of the chapter when they require more background than before. There's an awful lot in here; this was my course text for Math 242 and I used it as one of the texts in a reading class on number theory, and I still haven't read through all the chapters. It's a great example of a book in which the authors have tried and succeeded in bringing advanced material down to the undergraduate level. Some good historical notes, as any self-respecting number theory text should contain. Recommended highly.

[BB] The book is composed entirely of exercises leading the reader through all the elementary theorems of number theory. Can be tedious (you get to verify, say, Fermat's little theorem for maybe 5 different sets of numbers) but a good way to really work through the beginnings of the subject on one's own.

This is the classic, and Hardy is one of the great expository writers of mathematics. However, I remember that the last time I looked at this book it made no sense to me. If you like number theory you should probably at least look at it.

[PC] Oh, here I must fervently disagree (well, okay, maybe it didn't make sense to you at the time, but please go ahead and look again). I say that any student of mathematics should have this book on their shelf. Here's H/W's game: they explain number theory to people who can follow mathematical proofs but have no prior exposure to the subject or any advanced machinery whatsoever—hmm, maybe a little calculus at times, but not always. The one thing they do use is a little asymptotic growth notation, i.e., O, o, and the squiggly line, and for some reason they assume that people will know all about this without much comment. I seem to recall that one chapter towards the beginning is confusing because of this, and when I first bought the book it stymied me (I was sixteen at the time). But it's written so that you don't have to read it in order: they develop just enough theory about almost every branch of (elementary) number theory so that you can see interesting theorems proved. I have jumped around a lot, but over the years I think I've read almost every chapter. I really think it's the #1 “cultural enrichment” book for math students.

[PC] Recommended to me by none other than Professor Narasimhan himself, it's actually a very elementary and readable introduction to the classic theorems of analytic number theory: Chebyshev's Theorem, Bertrand's Postulate, uniform distribution, Dirichlet's Theorem and the Prime Number Theorem. Requires epsilonics and just a little bit of complex function theory.

[PC] If you've been reading this list, you know from Chris that Apostol writes terribly dry books. I've never read anything by him but this one, and it's fine, a bit more elementary than Chandrasekharan and easier to get your hands on (Apostol is a UTM; Chandrasekharan is an out of print Springer international edition). It starts out with a nice introduction to arithmetic functions, including the convolution product, and it covers much the same as the above, only a bit less briskly. A quick route to the proofs of the greatest theorems of 19th century mathematics.

The first chapter of Knuth's immortal work The art of computer programming is an extensive study of combinatorics and asymptotics. G/K/P is an expanded and friendlier version, which emphasizes teaching the reader to solve things, rather than just showing how they are done. Contains many funny marginal notes from students in the Stanford class which gave birth to the book, as well as tons of great exercises. Not a reference work.

The first eight chapters of this little book form the best, cleanest exposition of elementary real analysis I know of, although few UC readers will have much use for the chapter on Riemann-Stieltjes integration. Like Rudin's other books, it is broken into bite-size pieces, so you can prove every statement in the book on your own if you're self-studying. If that isn't enough, there is a large collection of challenging exercises. Some people think Rudin is too skinny and streamlined, but I think it's beautiful. (Ignore chapters 9 and 10, which are a confusing and insufficiently motivated development of multivariable calculus. Chapter 11 is all right for Lebesgue integration, but there are better treatments elsewhere.)

[PC] I agree 100% with what Chris says, but I want to add my voice that this is (through chapter 8) the cleanest exposition I have ever seen. I still flip back to this to check things out.

[BR] I must insist that Chapters 9 and 10 are not THAT bad. They're worth revisiting if you are tired of Spivak and do Carmo.

Covers the same material as Rudin, plus a little complex analysis. Apostol assumes (hence, engenders) less maturity on the reader's part, writing most arguments out in “advanced calculus” detail rather than “real analysis” detail, if that makes sense. I find it terribly dry. Nevertheless the book is careful and comprehensive, with many exercises.

When I started 207 I couldn't see why the material of this book was analysis: here was set theory, some linear algebra, some stuff about normed linear spaces, a little functional analysis... oh, here's that cool integral everyone talks about, but where are the derivatives? Now I know why it's analysis, of course, but the book as a whole is still a perplexing beast to the inexperienced. I think the primary reason it remains a text for 207 is that it costs $13, so why not? The style is distinctively Russian, which puts me off but turns other people on. Extended applications appear occasionally to lend context, but on the whole there is little motivation (and few exercises). The book is also difficult to use as a reference work, because the authors develop only the results they need to get where they're going.

[PC] Agreed. But it's cheap and though you may wonder why you're learning so much functional analysis before you see a Lebesgue integral, it's still clear and easy to read, so there's no reason why you shouldn't own it.

Covers the same material as K/F, with the addition of a chapter relating differentiation to Lebesgue integration (the fundamental theorems of calculus). H/S use the Daniell integral rather than K/F's concrete, bare-hands construction of Lebesgue measure; it's probably good to do it by hand once, but after that forget it. The sequence of topics makes a little more sense than K/F, although the chapter on inner product spaces is lonely at the end, where it lives because they want to do Fourier series. But the book is written in a ho-hum style, and the exercises are too easy. In this H/S shares the flaw of many books at this level, of making too big a deal of a little bit of abstraction which might be new to the reader. I went straight from little Rudin to big Rudin without much of a stop for either of these books.

This is an old, classic book which is worth a look. They develop many concrete classical topics (all those things like Legendre polynomials that you were always curious about) as exercises.

This book is a strange bird, the first volume of a nine(!)-volume treatise by one of the original Bourbakistes. I can't really describe it except to say that it's very formalistic, it has many good exercises, it's very hard to relate to other treatments of the subject, and it made a big impression on me.

The first half is the standard reference for real analysis (the second half is reviewed below). It's a very clean treatment of the topics it covers, again in bite-size pieces and with manychallenging exercises. Sometimes I get frustrated with the lack of motivation, or with Rudin's habit of proving exactly the lemmas he needs to do something, without any context for the results. Nevertheless it's a good reference or self-study book. Topics: Integration and L^p spaces, Banach and Hilbert spaces, Radon-Nikodym theorem and differentiation, Fubini's theorem, Fourier transforms.

[PC] Yes, how wonderful that there's one book whose first half contains all the analysis that you'll ever need to know! This book is advanced and the exposition is austere (“which gives (5). Applying (3) to (4), we get (6)”) but it is absolutely crystalline in its clarity (exception: is its proof of the L^2 inversion theorem for Fourier transforms valid? I'm not so sure.) Isn't this the one math book that every student must buy sooner or later (aside from Hardy and Wright, of course)? Some rainy day you'll discover that the book has a second half and find some very interesting theorems in there, but don't confuse it with a course on complex analysis, because it's a weird-ass treatment of complex analysis viewed through the eyes of a conventional analyst. Think of it as a bonus.

Another Serge Lang book, but a Serge Lang book is about the only place you'll find the inverse function theorem systematically treated for Banach spaces (except Dieudonné, and Lang was a Bourbakiste too).

Royden is like Hungerford for me: a lot of people like it, but it annoys me for a number of semi-silly reasons. He denotes the empty set by 0 (zero) and the zero element of a vector space by lowercase theta. He proves many theorems three times in gradually increasing generality. He leaves whole proofs to the exercises, and then depends on them later in the text. And I don't like his construction of Lebesgue integration. (Nyaah, so there.)

[BR] This is such a terrible book! He leaves the hardest theorems to the reader and proves some really simple-minded things with too much machinery. For example, he assigns the Urysohn lemma for normal spaces as an exercise for the reader and then has to use the Baire category theorem to show that on Banach spaces, linear operators are continuous iff they commute with taking limits. If you have to take 208 or 272, find a supplementary text. You'll be happy you did.

This is the book everybody gets in differentiation and integration in R^n, and it's a pretty good one, although the integration chapters are hard to read—maybe it was just my first encounter with exterior algebra that made it hard. As usual for Spivak books, clear exposition and lots of nice exercises. Unfortunately this one is old enough to be annoyingly typeset.

[PC] I don't really like this book, and I'm a big fan of Spivak in general. Does anybody else think that this rigorous multivariable Riemann integral theory is a dinosaur? And when Spivak starts talking about chains (in chapter four, I think), I don't know what the hell he's talking about. Presumably you could ignore that chapter and use the book as an introduction to differential forms. I can't suggest a substitute at the moment, other than Spivak's Comprehensive introduction volume 1, which is a wonderful book but which I still wouldn't want to read as a first introduction to forms. Come to think of it, I love forms to death, but maybe they're just plain confusing the first time around...

This skinny yellow book has replaced Munkres's Analysis on manifolds as the text for 274, and I'm not sure it's an improvement. It's more like a modernized Calculus on manifolds. I haven't done more than glance through it, but the notation is reputedly horrible, and Spivak is definitely a superior expositor.

Yes, Virginia, there is an interesting geometric theory of differential equations (of course!), not just the stuff you see in those engineering texts: stuff about stable and unstable points or manifolds, and other things with a dynamical-systems flavor. Nevertheless there is substantial material on how to reduce a differential equation to linear form and solve it, although no Laplace transform techniques or the like. Arnold explains it all coherently at an advanced-calculus level (manifolds appear at the end), complete with many beautiful diagrams. Another distinctively Russian book—read all the ones I describe that way, and you'll see what I mean. The third edition is substantially different from the second (which I have): the manifolds material is much expanded, and the typesetting is not so nice.

A tiny book which covers material similar to Arnold, but more concisely. I haven't read it but it's frequently referenced, and worth a look if you need to know the basic theorems. (If all you need is the basic existence-uniqueness theorem for ODEs, it's also in Spivak volume 1 or Lang, Real and functional analysis.)

Munkres's book is a wonderful first encounter with topology; in fact it begins slowly enough to be a first encounter with abstract mathematics (after a traditional advanced calculus course). Every abstraction is carefully motivated, and there are tons of examples, pictures, and exercises. This is one of those books you could hand to a bright student of any age who knew some calculus (not a bad book to choose if you're coming back to mathematics at age 35). Most of the book is the traditional analysis-topology material, but there is a long last chapter on the fundamental group which covers enough to prove the Jordan curve theorem.

[PC] Yes, Munkres deserves to be the standard undergraduate point-set book. It doesn't have everything, but it has most of the standard topics and it's relentlessly clear.

But Willard is my topology book of choice. The level of abstraction is deliberately higher, and the book is better organized as a reference than Munkres. It's not nearly as friendly, but it's still clear and well-written (I think an unclear point-set topology book is probably no longer a point-set topology book). Willard is probably the best modern reference for analysis-topology, where “modern” means “excluding Kelley” (see below). You can learn from it too; it's organized bite-size like a Rudin book, so you can prove all but the hard theorems on your own (I did this with an initial segment, and learned a lot).

[PS] Let me just say that Kelley's book on topology is horribly old-fashioned—I know because my advisor is forcing me to read it. Half the topics are things which I don't think are as important as they used to be. Nets, filters? I guess they're interesting in and of themselves. On the upside, it does have a nice appendix covering the rudiments of set theory.

[CJ] It is old-fashioned, but it's still the best book on topology for functional analysis, bar none. Nets are surprisingly necessary in infinite-dimensional topological vector spaces! The occasional proof is easier to read once recast in modern language, but doing so is a good learning exercise anyway. And Kelley has the nice habit (emulated less successfully by Willard) of treating substantial pieces of analysis as exercises; two of the exercises to Chapter 2 are titled “Integration theory, junior grade” and “Integration theory, utility grade”. It's really an analysis book disguised as a point-set topology book, but then much of functional analysis is really general topology on spaces that happen to be vector spaces too.

This is a topology ‘anticourse’: a collection of all the screwed-up topological spaces which provide limiting counterexamples to all those point-set topology theorems with complicated hypotheses. It's a classic just for the content, but pretty well written too. This book and Gelbaum/Olmsted (above) are two parts of what should someday be the big book of counterexamples to everything. Read it and see just what you avoid by sticking to differentiable manifolds.

[BR] Steen and Seebach have catalogued 143 of the most disgusting pathological topological creatures. They are invaluable for when you're first learning point set topology and need to understand why the definitions are necessary. They can also come in handy on tests: I used the one-point compactification of an uncountable discrete space three times on my Math 262 final. The text used for 262, Munkres, relies on three counterexamples to disprove everything: the Sorgenfrey line, S_Omega and I x I in the dictionary order. Steen and Seebach let you know that there are tons of other beastly topological spaces which violate the laws of common sense.

[YU]This is a point-set topology book. Less elementary than Munkres, but useful as a reference book for grad students.

I didn't understand transversality at all until I saw this book. It's a very geometric (as opposed to formalistic), down-to-earth introduction to some of the most mystical areas of smooth manifold theory: transversality and intersection theory. Abstraction is avoided (manifolds are defined as embedded in Euclidean space, which annoys me just a bit), but without hand-waving important distinctions (they are careful to point out that for noncompact manifolds, an injective immersion need not be an embedding, that is, proper too). The last chapter treats integration and Stokes's theorem, but that's not what anyone reads the book for. Beautifully written, and fills an important hole in Spivak volume 1.

We used this book for Corlette's differential geometry seminar two years ago (293). I didn't like it all that much because do Carmo is careful to keep the book to a post-advanced-calculus level: everything takes place in R^3, no vector bundles, lots of componentwise calculations. Nevertheless it's a nice treatment of the classical theory of curves and surfaces in space. Read it if you want to know about the Gauss map or the two fundamental forms, but don't want to work all the way through Spivak volume 2.

[PC] Volume 1 is the best introduction to smooth manifold theory and differential topology that I know of. Every chapter of this book has come in handy for me at one time or another. Ben and I like to describe the book as “locally readable”: his exposition is very careful, but sometimes he takes too damn long to explain a single concept. Luckily, despite Spivak's efforts to the contrary, you can flip around and read chapter by chapter, and I recommend this. There is so much good stuff in here.

[CJ] Buy it and read it over and over and over. Don't skip the exercises because that's where he puts all the freaky examples. It's true that sometimes he talks too much, but for the loving detail in which he lays out difficult concepts, he can be forgiven.

As Spivak puts it at the beginning, “Volume 1 dealt with the ‘differential’ part; in this volume we finally get down to some geometry.” Volume 2 treats the classical theory of curves and surfaces using the modern machinery developed in the first volume, which makes it (for me) a more comfortable read than do Carmo. Spivak is careful to motivate everything historically; surface theory is introduced by a long walk through Gauss's General investigations of curved surfaces (you should really have a copy of it to read this book), and the second half of the book goes through the (convoluted) stages of evolution of the definition of a connection. Not easy reading but every bit as rewarding as Volume 1. Unfortunately there are almost none of the wonderful exercises which characterize the first volume.

This is an interesting book which I can't really describe. It contains a number of short treatments of undeniably geometric but nontraditional topics; one fascinating application is the relation between phyllotaxis (the arrangement of plants' leaves around the stem) and generalized Fibonacci-type numbers. Read for culture.

The algebraic geometer of the famed book from hell (see below) recently finished another modern-Euclid book. I haven't seen it and don't even remember the title, but it might be interesting.