Metric spaces度量空间
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作者： Satish Shirali，Harkrishan L. Vasudeva著
出 版 社：
出版时间： 2005-8-1字数：版次： 1页数： 222印刷时间： 2008/08/01开本： 16开印次： 1纸张： 胶版纸I S B N ： 9781852339227包装： 平装内容简介
This volume provides a complete introduction to metric space theory for undergraduates。It covers the topology of metric spaces，continuity，connectedness，compactness and product spaces，and includes results such as the Tietze-Urysohn extension theorem，Picard's theorem on ordinary differential equations，and the set of discontinuities of the pointwise limit of a sequence of continuous functions。The key features include: a full chapter on product metric spaces，including a proof of Tychonoff's Theorem a wealth of examples and counter-examples from real analysis，sequence spaces and spaces of continuous functions numerous exercises - with solutions to most of them - to test understanding。The only prerequisite is a familiarity with the basics of real analysis：the authors take care to ensure that no prior knowledge of measure theory，Banach spaces or Hilbert spaces is assumed。The material is developed at a leisurely pace and applications of the theory are discussed throughout。
目录
0.Preliminaries.
0.1.Sets and Functions.
0.2.Relations.
0.3.The Real Number System
0.4.Sequences ofReal Numbers.
0.5.Limits of Functions and Continuous Functions
0.6.Sequences of Functions.
O.7.Compact Sets.
0.8.Derivative and Riemann Integral
0.9.Cantor’S Construction
0.10.Addition、Multiplication and Order in R.
O.11.Completeness ofR
1. Basic Concepts
1.1.Inequalities
1.2.Metric Spaces
1.3.Sequences in Metric Spaces
1.4.Cauchy Sequences
1.5.Completion of a Metric Space
1.6.Exercises
2.Topology of a Metric Space
2.1.Open and C：losed Sets
2.2.Relativisation and Subspaces
2.3.Countability Axioms and Separability
2.4.Baire’S Category Theorem
2.5.Exercises
3.Continuity.
3.1.Continuous Mappings’
3.2.Extension Theorems，
3.3.Real and Complex—valued Continuous Functions
3.4.Uniform Continuity
3.5.Homeomorphism，Equivalent Metrics and Isometrv
3.6.Uniform Convergence of Sequences of Functions：
3.7.Contraction Mappings and Applications
3.8.Exercises
4.Connected Spaces
4.1.Connectedness
4.2.Local Connectedness
4.3.Arcwise Connectedness
4.4.Exercises
5.Compact Spaces.
5.1.Bounded sets and Compactness
5.2.Other Characterisations of Compactness
5.3.Continuous Functions on Compact Spaces
5.4.Locally Compact Spaces一
5.5.Compact Sets in Special Metric Spaces
5.6.Exercises.
6.Product Spaces
6.1.Finite and Infinite Products of Sets
6.2.Finite Metric Products.
6.3.Infinite Metric Products
6.4.Cantor Set
6.5.Exercises
Index.
