It seems to me as if many modern texts are rather streamlined. They are designed not to expect too much from the reader but they often miss the depth of respective classical literature.
The purpose of this record is to collect highly recommended classical texts. Quality and depth of the subject matter should serve as a benchmark. Suitability for beginners should be irrelevant.
For a text to be regarded classical it should have proven its value for one decade at least. Nevertheless, it should expose the present state of the art.
The final list will contain each recommended text with $10$ or more votes.
In order to make this work, please restrict yourself to one proposal per answer/comment.
Thank you for your contribution!
Introduction to Commutative Algebra by Atiyah and MacDonald.
Complex Analysis by Lars Ahlfors.
Inequalities by G. H. Hardy, J. E. Littlewood and G. Pólya. (1934)
Gathers into one place techniques and results useful in many areas of mathematics.
Algebraic Number Theory, by Cassels and Frohlich.
Topology from the Differentiable Viewpoint, John Milnor.
Algebraic Geometry by Robin Hartshorne
Algebra, by Michael Artin.
Talking about classics:
Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press.
The first edition was published in 1922 and the second in 1944. Neverthless, after 90 years, Watson’s Treatise is still the standard reference book on the theory of Bessel functions (e.g., it has 10334 citations in Google Scholar).
Its latest reprint was released in 1996.
Elements of Mathematics, Nicolas Bourbaki.
Emil Artin, Galois Theory, Lectures Delivered at the University of Notre Dame.
Freely and legally available at Project Euclid.
I would say that the following are without a doubt considered some of the paradigmatic “classics”,
A Course in Arithmetic, Jean Pierre Serre.
Disquisitiones Arithmeticae, Carl Friedrich Gauss.
A French translation is available at Internet Archive and at Google Books:
$\bullet$ Internet Archive,
$\bullet$ Google Books.
Algebraic Topology, Allen Hatcher.
Linear Algebra and Its Applications, Peter Lax.
Morse Theory by John Milnor.
Differential Geometry of Curves and Surfaces by Manfredo Do Carmo.
Katznelson’s Introduction to Harmonic Analysis – for when Zygmund will break your shelf
Elementary Theory of Analytic Functions of One or Several Complex Variables, Henri Cartan.
Topology and Geometry by Glen Bredon.
This is a fairly recent book [1993, I think] but it’s a great book for a graduate algebraic topology course. It certainly isn’t easy-going, but there are pretty nice exercises at the end of each section. Additionally, the point-set section uses some nice notions [nets, for example] that the student may have missed out on in undergrad topology.
Personally, I was enlightened by
Here are some in Analysis…
Introduction to the Theory of Computation, Michael Sipser.
Books that I have learned a lot from that probably belong in the “classics” category are
Ok, the last one isn’t really a “classic”, per-se, since it’s only about 10 years old but it is a very good book.
Sorry for my definition of “classic” category.
I haven’t read the first one here at all, but it seems a favorite:
That’s easy-and I’ll focus on the books not already mentioned here.In no particular order of importance:
Theory of Functions by E. Titchmarsh
Lectures on Elementary Topology And Geometry by I.M. Singer and James Thorpe
Differential Topology by Victor Guillemin and Alan Pollack
Notes On Differential Geometry by Noel J.Hicks
General Topology by John Kelley
Differential Equations with Applications and Historical Notes by George F.Simmons
Elements of Differential Geometry by Richard Millman and Thomas Parker
General Theory Of Functions And Integration by Angus Taylor
Foundations of Differentiable Manifolds And Lie Groups by Frank Warner
Analysis And Solution Of Partial Differential Equations by Robert L. Street
Algebraic Topology by C.F.Maunder
Algebra by Roger Godement
Anything by either Einar Hille or Albert Wilansky
And those are just the ones I can think of off the top of my head. I’m sure I can come up with more.
A Concise Introduction to the Theory of Numbers, Alan Baker.
It seems to me that this book is not very popular, and I’ve never understood why.