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Title: Metric Spaces by V. W. Bryant ISBN: 0-521-31897-1 Publisher: Cambridge University Press Pub. Date: 01 June, 1985 Format: Paperback Volumes: 1 List Price(USD): $16.00 |
Average Customer Rating: 4.67 (3 reviews)
Rating: 5
Summary: Great Introduction to Metric Spaces. Lively, Informal Style
Comment: This short book is a gem. Metric Spaces is an especiallywell-written introduction to analysis. I liked the author's informalconversational approach to a rather abstract topic. Although I did find it necessary to reread substantial sections for full understanding, I found Metric Spaces to be quite enjoyable.
Dr. Bryant motivates the reader immediately with a look at iterative techniques, fixed point functions, converging sequences, and approximation solutions - all in an engaging style. Later topics included distance concepts, function spaces, and the relationship between closed sets, complete sets, and compact sets. The fourth chapter was devoted to the contraction mapping principle and its use in solving differential equations.
Is this book for you? The author says: "The only prerequisite is to have done a course on elementary analysis: it is not a prerequisite to have understood it nor to have remembered it at all." I personally had no formal courses in real or functional analysis and the highly structured axiomatic approach of analysis texts had never appealed to me. I only had a vague idea as to the properties of a metric space. But I was lured into buying Dr. Bryant's short text by the previous Amazon reviewers. And thankfully so.
Dr. Bryant clearly enjoys his subject, but he just as clearly recognizes that not everyone might have such an abiding interest. Throughout the text, he points out opportunities where the reader might skip forward if the going has become less interesting. (For the record I refused to be enticed by these short cuts.)
Problems are embedded in the text, one or two at a time, and are used to reinforce points under discussion. Most have clear hints and I found many problems straightforward, but others were more difficult. A few problems were identified as appropriate for the "keen" student.
The most abstract mathematics are reserved for the last (optional) chapter, but the author does encourage the reader to stay with it: "It would be a pity to stop ..." Chapter five recasts the first four chapters into a more generalized form of real analysis and addresses the question: "What makes analysis work?"
Dr. Bryant had an unusual goal for a mathematics text. "I have tried to provide a readable and natural introduction to an abstract subject in a down-to-earth manner." Also, he says, "My aim is to provide a book which can be read and enjoyed ..." He succeeded in doing just that.
Rating: 5
Summary: Metric Spaces
Comment: A short but excellent book for someone who wants a well motivated refresher on analysis. By grounding the ideas with applications of fixed point theorems (such as proving the existence of unique solutions to certain types of differential equations) the author makes accessible an area of mathematics that is often treated in an axiomatic and uninteresting way.
I believe the author is correct when he recommends the book for people who have already had some exposure to analysis. At best a student should already have completed the standard non-rigorous college calculus sequence to get the most out of this book.
Rating: 4
Summary: This is a great introduction to fixed-point methods.
Comment: This little book can take a bright high school student or undergraduate through some most interesting territory, previously uncharted at this level of mathematical understanding. Fixed point methods -- approaches to problem solving based on the simple equation f(x)=x -- are ubiquitous and ideally suited for digital computers, but they are seldom introduced in the classroom. Get this book for your high school library. Use it as an independent study for that student you don't want to set loose on calculus quite yet.
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Title: Yet Another Introduction to Analysis by Victor Bryant ISBN: 052138835X Publisher: Cambridge University Press Pub. Date: 01 November, 1990 List Price(USD): $33.00 |
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Title: Counterexamples in Analysis (Dover Books on Mathematics) by Bernard R. Gelbaum, John M. H. Olmsted ISBN: 0486428753 Publisher: Dover Publications Pub. Date: 01 June, 2003 List Price(USD): $14.95 |
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Title: Introduction to Metric and Topological Spaces by Wilson A. Sutherland ISBN: 0198531613 Publisher: Oxford University Press Pub. Date: 01 June, 1975 List Price(USD): $39.50 |
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Title: General Theory of Functions and Integration by Angus E. Taylor ISBN: 0486649881 Publisher: Dover Publications Pub. Date: 01 March, 1986 List Price(USD): $19.95 |
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Title: Div, Grad, Curl, and All That: An Informal Text on Vector Calculus by H. M. Schey ISBN: 0393969975 Publisher: W. W. Norton & Company Pub. Date: 01 October, 1996 List Price(USD): $17.95 |
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