Are there any books that present theorems as problems? To be more specific, a book on elementary group theory might have written: "Theorem: Each group has exactly one identity" and then show a proof or leave it as an exercise. The type of book that I am imagining would have written "Problem: How many unit elements can a group have?" and similarly for all other theorems.
Algebraic Geometry by Robin Hartshorne. An algebra professor once told me that almost every exercise is a lemma or theorem from SGA.
A friendlier and more accessible book for undergraduates is "Linear Algebra Problem Book" by Paul R. Halmos. Halmos is an awesome expositor and this one is no exception.
Some classical books that would probably fit the bill:
Problems and Theorems in Analysis by Polya and Szego
A Hilbert Space Problem Book by Halmos
Fifty challenging problems in probability with solutions by Frederick Mosteller. It deserves to be better known than it is. Some things I like about it:
- It is elementary enough to be readable by high school students, but it introduces some serious ideas of probability.
- It is entertaining!
- It sells for $6.95.
My favorite such book is Problems in Analytic Number Theory by Ram Murty. There could not be enough good things said about it.
A number of books by Russian authors (cf. the one by Viro et al. from the Anton's answer) also come close to what you ask for. The two that came to my mind first are:
Theorems and problems in functional analysis by Kirillov and Gvishiani
Elementary Topology. Textbook in Problems by O.Ya.Viro, O.A.Ivanov, V.M.Kharlamov, N.Y.Netsvetaev
Here is my favorite: Victor Prasolov, Problems and Theorems in Linear Algebra. Also avaliable in PDF (mirror). However, I wouldn't recommend this to any undergrad without olympiad background.
Generally, writing textbooks in form of problem compendiums is distinctive for Soviet mathematics. I could name some more books of this kind (such as one on Lie algebras), but unfortunately they are all in Russian and most have never been translated.
Pinter's A Book of Abstract Algebra is half problems, half text. Many important topics are covered as problems. For example, direct products of groups are introduced and their properties developed in a set of problems. Cauchy's Theorem and Sylow's Theorem are introduced as problems. I taught myself a good deal of abstract algebra from this book one summer. The high proportion of problems to exposition kept me stimulated, and his decomposition of proofs of theorems into bite-sized problem chunks was a confidence booster. I remember this book with great affection.
"Elements of the Theory of Representations" by A. Kirillov. This is a concise introduction to the representation theory of both finite and Lie groups. It contains necessary background from other fields, e.g. analysis on manifolds. Many theorems are formulated as problems, often with hints. Originally the book was written in Russian, but there is also English translation published by Springer-Verlag in 1976.
Convex Figures I.M. Yaglom and V.G. Boltyanskii Holt, Rinehart and Winston, NY, 1961
The first half of this book has definitions and results related to convexity to be proved by the reader and the solutions to these problems (theorems) is given in the second half of the book.
The topics treated include Helly's Theorem, isoperimetric results, Minkowski addition of sets and curves of constant width.
All of this material is clearly and well handled.
I had a lot of fun skimming through Jim Henle's An Outline of Set Theory. (It's now out of print, but I suspect Springer's Problem Books in Mathematics series has a few more such titles still in print.)
Also, a nice little freebie is Stefan Bilaniuk's A Problem Course in Mathematical Logic.
In a different area, there is Number Theory Through Inquiry by David C. Marshall, Michael Starbird, Edward Odell.
Ian Adamson has 2 really nice books pitched at the upper level undergraduate/graduate level: A General Topology Workbook and A Set Theory Workbook. Set Theory and point set topology can mostly be developed directly from the definitions,so these are nice subjects to present in this manner,particularly to students just learning how to do rigorous proofs. Best of all,both books come with complete solutions in the back.
Kenneth P. Bogart's "Combinatorics through Guided Discovery" is written exactly this way and all properties touched on in the book are discovered in the book through problems.
You can download it here: http://www.math.dartmouth.edu/news-resources/electronic/kpbogart/
The little commutative algebra book by Atiyah and MacDonald is one such—the reason it's so little is that probably two-thirds of the results in it are in the exercises. I guess you know the subject if you can skim through it looking for a fact (which is quite likely to be present, if not in the main text) all the while nodding your head as though the proofs were actually given.
An introduction to the theory of groups by Joseph J Rotman makes for a good DIY second course in group theory.
Learning mathematics by solving problems is part of the french tradition. You will find many problems in Bourbaki or Dieudonne's Elements d'Analyse. At a more elementary level there are several problems covering a large amount of material at the end of Colmez's Elements d'analyse et d'algebre (et de theorie des nombres).
There are several introductory problem texts by R.P. Burns:
A nice introduction (highly recommended!) to abstract algebra which is almost entirely presented in problem format is the following by Clark:
I recently found the following books (which I have no experience with):
"Selected Problems in Real Analysis" by M. G. Goluzina, A. A. Lodkin, B. M. Makarov, A. N. Podkorytov. (The authors are modest with the title, I would rather say that this book contains all problems in real analysis.) This is a perfect complement to Polya and Szego, whose book is rather on complex analysis.
There are number of books in the Schaum's Outline series that I would recommend to anyone beginning in the subject of choice, Group Theory, Linear Algebra, General Topology to name a few. They are good in making the initial learning curve less steep, and help to make many of the other books mentioned more accessible to someone new to the subject.
Gerhard "Ask Me About System Design" Paseman, 2012.05.13
I would like to mention about Serge Lang's Algebra. Many of the "standard" results/theorems appear as exercises: one particular example is the construction and properties of Witt ring as exercises in chapter XV; of course, there is "that famous" homological algebra problem too.
See these two good books :
Combinatorics : a problem oriented approach - Daniel A. Marcus
Graph Theory : a problem oriented approach - Daniel A. Marcus
The first one was recommended for self-study by an AMS article :
Marcus’s elementary Combinatorics: A Problem Oriented Approach is appropriate for self- study; more advanced is Lovász’s Combinatorial Problems and Exercises.
See also :
Some answers mention problem books (quite different from standard-format textbooks in that they consist almost entirely of problems and their solutions). Such books have been widely used in Eastern Europe at every level of education (at least when I was getting it). Let me add another one to the list:
MR0447533 Krzyż, Jan G. Problems in complex variable theory. Translation of the 1962 Polish original. Modern Analytic and Computational Methods in Science and Mathematics, No. 36. American Elsevier Publishing Co., Inc., New York; PWN---Polish Scientific Publishers, Warsaw, 1971. xvii+283 pp.
In his foreword, the author states: ``Most exercises are just examples illustrating basic concepts and theorems, some are standard theorems contained in most textbooks. However, the author does believe that the reconstruction of certain proofs could be instructive and is possible for an average mathematics student."
Besides standard material, there is a collection of quirky little facts in e.g. non-Euclidean geometry in the disk or logarithmic potential theory (and much more). All stated as problems for the reader to solve. However, many solutions are included.
There were subsequent editions in Polish. I used one as an undergraduate student and still have a copy.