Metric Spaces

ISBN-13: 9781846283697 / Angielski / Miękka / 2006 / 304 str.

Mícheál O'Searcoid

Opis

The abstract concepts of metric spaces are often perceived as difficult. This book offers a unique approach to the subject which gives readers the advantage of a new perspective on ideas familiar from the analysis of a real line. Rather than passing quickly from the definition of a metric to the more abstract concepts of convergence and continuity, the author takes the concrete notion of distance as far as possible, illustrating the text with examples and naturally arising questions. Attention to detail at this stage is designed to prepare the reader to understand the more abstract ideas with relative ease. The book goes on to provide a thorough exposition of all the standard necessary results of the theory and, in addition, includes selected topics not normally found in introductory books, such as: the Tietze Extension Theorem; the Hausdorff metric and its completeness; and the existence of curves of minimum length. Other features include:

• end-of-chapter summaries and numerous exercises to reinforce what has been learnt;
• extensive cross-referencing to help the reader follow arguments;
• a Cumulative Reference Chart, showing the dependencies throughout the book on a section-by-section basis as an aid to course design.

The book is designed for third- and fourth-year undergraduates and beginning graduates. Readers should have some practical knowledge of differential and integral calculus and have completed a first course in real analysis. With its many examples, careful illustrations, and full solutions to selected exercises, this book provides a gentle introduction that is ideal for self-study and an excellent preparation for applications.

Recenzje

From the reviews:

"This book is truly about metric spaces. ... The book is packed full of material which does not often appear in comparable books. ... His use of questions to increase understanding and to move on to the next topic are also to be appreciated. ... this is a great book and suitable ... for third-and fourth-year under-graduates and beginning graduate students." (Marion Cohen, MathDL, January, 2007)

"The book is very readable. It includes appendixes on the necessary mathematical logic and set theory, and has a substantial number of exercises... Every concept is demonstrated via a large number of examples, starting with commonplace ones and expanding the reader's horizon with the more abstruse ones, to give a sense of the scope of the concepts... A useful addition to any library supporting an undergraduate mathematics major." (D. Z. Spicer, CHOICE, March, 2007)

Spis treści

To the Reader.- Cumulative Reference Chart.- Metrics.- Distance.- Boundary.- Open, Closed and Dense Sets.- Balls.- Convergence.- Bounds.- Continuity.- Uniform Continuity.- Completeness.- Connectedness.- Compactness.- Equivalence.- Appendices: Language and Logic.- Sets.- Solutions.- List of Symbols.- Index.

O autorze

Mícheál Ó Searcóid is author of another, higher-level, SUMS book, Elements of Abstract Analysis, 1-85233-424-X, published November 2001, sales (as of June 2005): 1051 (ROW: 634; US: 417).

Blurb

The abstract concepts of metric ces are often perceived as difficult. This book offers a unique approach to the subject which gives readers the advantage of a new perspective familiar from the analysis of a real line. Rather than passing quickly from the definition of a metric to the more abstract concepts of convergence and continuity, the author takes the concrete notion of distance as far as possible, illustrating the text with examples and naturally arising questions. Attention to detail at this stage is designed to prepare the reader to understand the more abstract ideas with relative ease.

The book goes on to provide a thorough exposition of all the standard necessary results of the theory and, in addition, includes selected topics not normally found in introductory books, such as: the Tietze Extension Theorem; the Hausdorff metric and its completeness; and the existence of curves of minimum length. Other features include:

• end-of-chapter summaries and numerous exercises to reinforce what has been learnt;
• extensive cross-referencing to help the reader follow arguments;
• a Cumulative Reference Chart, showing the dependencies throughout the book on a section-by-section basis as an aid to course design.

The book is designed for third- and fourth-year undergraduates and beginning graduates. Readers should have some practical knowledge of differential and integral calculus and have completed a first course in real analysis. With its many examples, careful illustrations, and full solutions to selected exercises, this book provides a gentle introduction that is ideal for self-study and an excellent preparation for applications.