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Reviews Written by Michael Greinecker (Vienna, Austria)







11 of 11 people found the following review helpful
The wonderful World of Mathematical Analysis, August 4, 2005
This guided tour through mathematical analysis and its foundations is quite different from all existing textbooks out there. The book emphazises the interconnections between different areas of mathematics. Apart from all the standard material found in all real analysis textbooks, it covers in detail material from algebra, logic and set theory, as it relates to analysis. The book really makes one see the larger picture.
It also allows one to draw finer distinctions. The book gives a great overview of various weak forms of the axiom of choice and their relative strength. It introduces the reader to other approaches to analysis like constructivism or nonstandard analysis. Questions that are usually glossed over are treated with care, omissions aren't covered.
Obviously, at about 800 pages, the book cannot treat every topic in depth. But even here the book excels. Equivalent and inequivalent definitions of concepts are presented and the reader can easily supplement his reading with other books. Where the way stops, it shows how to go further.
The book requires a good deal of mathematical maturity. The reader has to fill in many holes in the proofs, but they are still manageable. The book isn't harder to read than conventional graduate level textbooks, but the reader will profit much more. The exercises are well integratet and make the reader actually do them.









12 of 13 people found the following review helpful
It's about Linear Algebra Done Right, stupid!, July 31, 2005
Linear maps from a finite dimensional vector space to another finite dimensional vector space can be represented by a matrix. Most books on linear algebra deal more with the representation than with the real stuff. But this is focusing on the map instead of focusing on the territory. Focusing on matrices brings a lot of notational cluster, leads to combinatorial and unalgebraic proofs and makes the transition to abstract algebra and functional analysis very hard.
This book avoids all these problems by focusing on the real deal. While the degree of abstraction is obviously higher, the proofs are much easier to understand and the text is more readable.
The book is written for a second course in linear algebra for math majors. If you aren't in the target group, don't blame it on the book. If you are math major, you shouldn't need any solutions for figuring out wether your proofs are correct or not. Things proved only in the exercises aren't used in the main text, so if you stumble eventually you can still read the rest of the book.









40 of 40 people found the following review helpful
Close to Perfect, April 12, 2004
This is pretty much the perfect introduction to set theory for someone having some familiarity with rigorous mathematics. The treatment is axiomatic but doesn't employ the usual logical formalism, everything is written in plain english. The book emphasizes the foundational character of set theory and shows how all the usual objects of mathematics can be developed using only sets. It also demonstrates the application of set theoretic methods to "ordinary" mathematics by giving complete proofs of some powerful theorems like the HahnBanach theorem in functional analysis. The pace is leisurely with a close look at the details. The axiom of choice is used only when necessary and it's uses are highlighted. The exercises contain real meat but are broken up in handable pieces. They also give alternative approaches to topics treated in the main text. Solutions are not contained. The last section is devoted to an outlook at more advanced set theory. The ideas of the constructible universe and of forcing are outlined, as far as that is possible on that level. There is also a discussion on candidates for additional axioms. The reader will gain both insight into what set theory is and how powerful it is. There is no better book for the same audience.









24 of 24 people found the following review helpful
Sen  But not at it's best, March 4, 2004
This short book consists of three lectures. In the first lecture Sen mainly argues that economic agents shouldn't be modelled as completely self centered. He argues that real individuals don't behave that way. I think that's fairly obvious and the reason people have often be modelled as egoistic is technical convenience. Other than that, he argues that rationality should mean more than consistent actions. Point taken. Since more work is done today that isn't based on selfcentered individuals, the lecture is of somewhat minor importance. But if you think all people act in an egoistic fashion (at least in an economic environment), read that part. The second lecture is IMO the best one. Sen looks on the impact utilitarianism had on economics at identifies the parts utilitarianism is made of. He the goes on to argue that these parts are independent and that their merit should be judged independently. He shows several roads one can take without accepting utilitarianism in its totality. In the third lecture Sen takes a look at the proper scope of social choice theory and things that should be incorporated but aren't yet. It's fairly good but nothing spectacular. Apart from the rough outline given, the worth of the book lies in little remarks Sen makes on a number of topics. These remarks make one think and reconsider ones position.









4 of 5 people found the following review helpful
A superb Introduction to Mathematical Economics!, February 6, 2004
On little less than 300 pages, Hildenbrand and Kirman give an exceptionally deep introduction to equilibrium analysis. The main topics are Walrasian equilibria (optimal choices relative to a price induced budget set), the core (a generalization of the familiar contract curve), and the relationship between the two. Everything is done in an exchange framework (i.e. no production), which makes it easier to focus on the concepts involved. Kirman and Hildenbrand prove the existence of Walrasian equilibria, give conditions for the core to be nonempty, show how the core of finite economies converges to the Walrasian equilibria and demonstrate the equivalency between Walrasian equilibria and the core in economies with an continuum of traders. In the last chapter they look at the structure of excess demand functions, conditions for uniqueness and stability in respect to tâtonnement dynamics (no trade until equilibrium is reached). Four appendices give essentially all the mathemtical background needed, except for a little calculus and linear algebra needed in the last chapter. A motivating overview of the content is given in an informal introduction. All results are well discussed and put in the proper context. The authors don't shy away from showing the limitations of the models. They also point out how one can weaken certain assumptions involved.









28 of 28 people found the following review helpful
The canonical Book on General Equilibrium Analysis, February 5, 2004
This short book (roughly 100 pages) gives a clear exposition of the basic elements of axiomatic general equilibrium analysis. The first chapter introduces all (sic!) mathematics used in this book, mainly some topology of euclidean space and basic facts about convex sets. In principle only knowledge about counting is necessary, but some "mathematical maturity" is clearly required. I would advise the reader to learn the relevant topology elsewhere ("Introduction to Analysis" by M. Rosenlicht suffices) and use the first chapter only for reference. The main text covers the ArrowDebreuMcKenziemodel and its interpretation, proves its logcial consistency (existence) and investigates its efficiency properties. The formal model is clearly distinguished from its interpretation, which allows Debreu to introduce uncertainty in the model by a simple reinterpretation of the commodity space. The whole approach is axiomatic, which wasn't that usual when Debreu wrote the book in 1959. This book has changed the standards of mathematical rigor in economic theory. This book is still used as a reference and deserves a place on every economic theorists bookshelf.









1 of 1 people found the following review helpful
All the GE that is fit to print..., February 5, 2004
This book contains the entries on General Equilibrium Theory from the New Palgrave Dictionary of Economics. The entries are short essays by the contributors on various topics, usually in the form of small surveys on a specific topic. The list of contributors reads like a WhoisWho of GET (Debreu, Scarf, McKenzie, Smale, Hildenbrand, Radner...) Pretty much every relevant topic is covered, from general surveys of GE to specific questions like the role of the freedisposalassumption in existence proofs. The essays are well written enough to be read for pure enjoyment, but equally useful as a starting point for the investigation of the various questions. Some essays are a little bit technical, but that's to be expected, given the subject field. At least grad students should be able to understand all of it.









3 of 4 people found the following review helpful
Keynes meets Walras, January 28, 2004
The concern of the book are the various approaches to unify macroeconomics and monetary theory with general equilibrium theory. The book consists of three parts. In the first part, the problems are outlined, the interpretations of Keynes and the standard ArrowDebreuMcKenzie model are survied, together with early approaches to unify them. The second part takes a look at Walrasian approaches, where prices and markets are central, and Edgeworthian approaches, where direct exchanges are examined, to macroeconomic questions. Both are examined from a static equilibrium perspective and dynamic disequilibrium theory. The third part is dedicated to reflections. The book is surveying the field, it is not a textbook. The discussion remains on an informal level, with little mathematics being used. The book should be readable to advanced undergraduates and more advanced readers. The only drawback is that the book is a little bit old. There is no discussion of OLG models, which became popular later. Since the topic of dynamics has been neglicted by GE theorists for quite some time, the book is still worth reading.









22 of 24 people found the following review helpful
Essential to understanding HET, October 13, 2003
E. Roy Weintraub investigates the relationship between the development of mathematics and economics. He argues that by ignoring that mathematics too is a changing field, historians of economic thought have missed important distinctions. In clarifying the strange relationship between Marshall and mathematical methods in economics he shows how this distinctions give new, important insights. He traces the story of the mathematician Griffith C. Evans and his attempt to do mathematical economics like physics with quantifyable data (influenced by Volterra). In his next chapter he looks at Hilberts influence in mathematics, which is distinct from his impact on metamathematics. Having set the stage for abstract formalisms, he investigates how Gerard Debreu has brought the views of Nicolas Bourbaki, a important abstractionist movement, into economics. The following two chapters aim to clarify the differences between mathematical and economic culture. As an illustration, he gives a account of a unfruitful correspondence between Don Patinkin and the eccentric mathematician, Cecil Phipps, who also was influencial in the puplication of the famous existence proof of Arrow and Debreu. After this, Weintraub get's personal and tells the story of his economist father and mathematician uncle and how economics become a topic for well trained mathematicians. Weintraub also tells his own story of a economist turned mathematician as a example of a large inflow of mathematicians into economics. The last chapter is dedicated to methodological issues.









18 of 19 people found the following review helpful
Good stuff!, September 23, 2003
The book is based on lecture notes of a course for Stanford CS students. The lecture notes aren't very polished, but the chatty tone makes good reading. The book is incredibly funny, but some people may not like that. Guest lecturers include outstanding people like Leslie Lamport and Paul Halmos. The content is excellent, especially the parts about writing proofs. Some parts are of more interest to computer scientists, but most of it is valuable to anyone engaged in mathematical writing. Despite all the good things, this book doesn't really stand alone and should be complemented with other books, like the one by Krantz. P.S.: A almost complete TEX version of the book can be found on the web.


