Metric Spaces

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Springer Science & Business Media, 26.12.2006 - 304 Seiten
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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.

 

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Inhalt

Metrics
1
Distance
21
Boundary
35
Open Closed and Dense Subsets
53
Balls 71
70
Convergence
83
Bounds
103
Continuity
125
Functions
262
Restrictions Extensions and Compositions
263
Mappings
264
Chains
266
Equivalence Relations
267
Sequences
268
Infinite Selection
269
Algebraic Structures
272

Uniform Continuity
147
Completeness
165
Connectedness
191
Compactness
205
Equivalence
227
Appendix A Language and Logic
245
Appendix B Sets
251
Unions and Intersections of Arbitrary Collections
261
Isomorphism
275
FiniteDimensional Linear Spaces
277
Solutions
279
List of Symbols 293
292
Bibliography
295
Index
297
Urheberrecht

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Über den Autor (2006)

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).

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