What are some classic books in mathematics?

Question

In his book "Men of Mathematics", Eric Temple Bell repeatedly makes the point that a student of mathematics must read the classics.

My question is what are some classic books in mathematics ( Dictionary definition : judged over a period of time to be of the highest quality and outstanding of its kind.) that can be used by a high school/undergraduate student to start the study of higher mathematics?

Some subjects I would like reference in particular, otherwise state any book you consider a classic, are:

1) Analysis 2) Abstract algebra 3) Linear Algebra 4) Number theory 5) Combinatorics and Graph theory, etc.

Answer 1

It's a bit more advanced than the topics you asked about, but Milnor's Morse Theory and Milnor and Stasheff's Characteristic Classes are astoundingly good. (There's a pattern here: Milnor's Lectures on the h-Cobordism Theorem is pretty good too!)

At a somewhat lower level, I find Spivak's Calculus (which many might argue is an introductory analysis book) pretty darned wonderful.

Answer 2

The Mathematical Association of America (MAA) has got a rich collection of classic books under Doclani Mathematical Expositions. I would suggest you following:

$1$. Mathematical Gems Series ($3$ Volumes) By Ross Honsburger.

$2$. Linear Algebra problem book By Paul R Halmos.

$3$. Euler: Master of us all By William Dunham.

Some other texts:

$1$ Introduction to Commutative Algebra by Atiyah and MacDonald.

$2$ A book of abstract algebra by Pinter.

Answer 3

1. Feller's first volume for Probability Theory.
2. Arnold's ODE for differential equations.
3. Cartan's Elementary Theory of Analytic Functions of One or Several Complex Variables for Complex Analysis

Answer 4

In the early '70s, I used two teaching books that I consider ''classic'':

Foundations of modern analysis of J. Dieudonné (at least in Europe).

Algebra of S. Mac Lane and G. Birkoff

At a different level, I think that an ''evergreen'' is:

Methods of Mathematical physics of R. Courant and D. Hilbert.

Answer 5

I don’t agree with Bell on this point: one may well learn better and more easily from a book that is not generally considered a classic. For example, most people have never even heard of John Greever’s modified Moore method textbook Theory and Examples of Point-Set Topology, but for me it was the ideal introduction to the field. That said, I can nevertheless name a few examples.

For someone of my generation I.N. Herstein’s Topics in Algebra is a classic introduction to abstract algebra. The first volume of William J. LeVeque’s two-volume Topics in Number Theory is a classic at the higher end of the undergraduate level; Underwood Dudley’s Elementary Number Theory is a classic at the lower end.