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I'm often impressed that top mathematicians in a given field seem to have not only a knowledge of the "state of the art" of their subfield, but also a knowledge of the history of the field and thus the seminal books/papers in the field.

In this direction, I would love to have a list of classic math texts (books, especially) that rate today not as mere historical curiosities, but that would be of benefit to a graduate student to read as a first introduction to a given field. Thus in some sense I'm asking for classic books that have not been rivaled or replaced. Are there such books? I'm told that Weyl's The Classical Groups is such a book. Are there others?

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While Weyl's book is amazing, "immediate practical benefit" is a strange phrase to put beside it... –  Mariano Suárez-Alvarez Jun 22 '12 at 19:32
    
Would you explain this comment in more depth? I was just about to start reading Weyl's book this summer. Maybe the phrase "immediate practical benefit" is not quite right. I mean to ask for books that give practical insight, not merely historical insight. –  Wouter Zeldenthuis Jun 22 '12 at 20:03
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Well, well, this is a dangerous question.

I'll venture to post an answer, nonetheless.

After reading your question for the first time, I thought I'd be able to name many books. But after thinking about it for a while, my personal (very subjective) list of books qualifying esp for 'classics' which (still) may serve as 'a first introduction to a given field' to a 'graduate student' for now reduces to

  • Richard Courant's 'Dirichlet's Principle, Conformal Mapping, and Minimal Surfaces'
  • J. Milnor/ J.T. Stasheff: 'Characteristic Classes'
  • J. Milnor: 'Morse theory'
  • David Gilbarg, Neil S. Trudinger: 'Elliptic Partial Differential Equations of Second Order'
  • Walter Rudin's 'Functional analysis'
  • Ingrid Daubechies: 'Ten Lectures on Wavelets'

with the title leaving no doubt which field might be addressed in each of the books. I'm particularly unhappy to recognize that I'm not able to add a book dedicated to differential geometry to the list.

Some of these may not yet be considered classics.

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(I'm sure someone is already planning to very slowly skin me cause I did not mention xyz but rather preferred to mention abc...;-) –  user20266 Jun 22 '12 at 20:13
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