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It is ultimately a matter of personal taste, but I prefer to see a long explicit example, before jumping into the usual definition-theorem path (hopefully I am not the only one here). My problem is that a lot of math books lack the motivating examples or only provide very contrived ones stuck in between pages of definitions and theorems. Reading such books becomes a huge chore for me, even in areas in which I am interested. Besides I am certain no mathematical field was invented by someone coming up with a definition out of thin air and proving theorems with it (that is to say I know the good motivating examples are out there).

Can anyone recommend some graduate level books where the presentation is well-motivated with explicit examples. Any area will do, but the more abstract the field is, the better. I am sure there are tons of combinatorics books that match my description, but I am curious about the "heavier" fields. I don't want this to turn into discussion about the merits of this approach to math (i know Grothendieck would disapprove), just want to learn the names of some more books to take a look at them.

Please post one book per answer so other people can vote on it alone. I will start:

Fourier Analysis on Finite Groups and Applications by Terras

PS. this is a similar thread, but the main question is different. How to sufficiently motivate organization of proofs in math books

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Great question. I look forward to seeing the responses. – John D. Cook Dec 6 '09 at 13:19

34 Answers 34

O. Ya. Viro, O. A. Ivanov, N. Yu. Netsvetaev, V. M. Kharlamov, Elementary Topology Problem Textbook has lots of examples.

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Usually old books solve this problem. For example books of Euler consist mostly of examples. It is interesting to read Jacobi, Weierstrass, etc. Maybe universal recipe is to start with a book 50-100 years old.

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No probability book yet, so let me add a classic.

William Feller, An Introduction to Probability Theory and Its Applications, vol I, II. Full of examples, well motivated.

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I can give a couple of dozen examples-but for now,I'll just list my favorite for topology/geometry: The trilogy by John M.Lee is probably the best written,laid out and flat out wonderful introduction to the study of differential and Riemannian manifolds there is for anyone looking to learn it on thier own. I hate to say it,but it's better then Spivak's opus.

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