Publisher: Chapman & Hall/CRC (2009)Details: 383 pages, Hardcover
Series: Pure and Applied Mathematics 293
Price: $89.95
ISBN: 9781584887935
Category: Monograph
Topics: Descriptive Set Theory, Logic, Set Theory
Marian Mureşan
Publisher: Chapman & Hall/CRC (2009)
Details: 383 pages, Hardcover
Series: Pure and Applied Mathematics 293
Price: $89.95
ISBN: 9781584887935
Category: Monograph
Topics: Descriptive Set Theory, Logic, Set Theory
[Reviewed by Allen Stenger, on 03/16/2009]
This book has a lot of weaknesses, and it’s hard to recommend it for anything. The most serious problem is that it is a hodge-podge and lacks focus. A large part of the book is a traditional course in real (not classical) analysis, similar to but less comprehensive than Apostol’s Mathematical Analysis (Addison-Wesley, 2nd ed. 1974) or Rudin’s Principles of Mathematical Analysis. Another large part of the book is a collection of specific results about particular series, functions, and constants; this is probably what the “classical analysis” in the title refers to.
The book starts out with two chapters on topology and abstract spaces (including Hilbert spaces but not Banach spaces), but most of this material is not used later. Metric spaces are used in many places, but not consistently; the exposition jumps back and forth between metric spaces and the real line without any explanation. The title term “Concrete” is explained in the Preface but I did not understand the explanation; it might mean “applied”, but there is not any applied math in the book.
Some stated results are simply wrong. The book claims on p. 17 that completeness of the real numbers is equivalent to the Archimedes principle. It claims on p. 209 that if the kth derivative of a Ck function has at least k+1 zeroes in an interval, then the function has a zero in the interval.
The exposition needs work too, although most of this could be fixed by better copyediting. The English is often unidiomatic (many theorems start with “It holds” or “It holds that”). Often terms are used many pages before they are defined. A number of theorems are stated without proof, while others are stated with proof but the proof is delayed for several or many pages, and there’s usually no indication of whether a proof will follow or not.
Classical analysis (which I define as “the kind that has numbers in it”) seems to be out of fashion today. My favorite classical analysis text, which has been out of print for many years, is Karl R. Stromberg’s An Introduction to Classical Real Analysis (Wadsworth, 1981). You can also find good coverage of this topic in many older books with “Theory of Functions” in the title, such as Titchmarsh’s Theory of Functions.
Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at MathNerds.com, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.