"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 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.
Author(s): Mícheál Ó Searcóid
Series: Springer Undergraduate Mathematics Series
Edition: 1
Publisher: Springer
Year: 2006
Language: English
Pages: 301
Tags: Математика;Функциональный анализ;
Metric Spaces......Page 4
Cover......Page 1
Front matter......Page 2
Contents......Page 7
1. Metrics......Page 20
2. Distance......Page 40
3. Boundary......Page 54
4. Open, Closed and Dense Subsets......Page 71
5. Balls......Page 88
6. Convergence......Page 100
7. Bounds......Page 120
8. Continuity......Page 142
9. Uniform Continuity......Page 164
10. Completeness......Page 181
11. Connectedness......Page 207
12. Compactness......Page 221
13. Equivalence......Page 243
Appendix A. Language and Logic......Page 261
Appendix B. Sets......Page 267
Solutions......Page 295
List of Symbols......Page 308
Bibliography......Page 310
Index......Page 311
