Metric Spaces (Google eBook)

Frontcover
Springer, 26.12.2006 - 324 Seiten
1 Rezension
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. The book provides 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. 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
11 Metric Spaces Definition
2
12 Point Functions and Pointlike Functions
8
13 Metric Subspaces and Metric Superspaces
10
14 Isometries
11
15 Extending a Metric Space
12
16 Metrics on Products
13
17 Metrics and Norms on Linear Spaces
16
93 Uniform Continuity on Subsets of the Cantor Set
153
94 Lipschitz Functions
154
95 Differentiable Lipschitz Functions
156
96 Uniform and Lipschitz Continuity of Compositions
157
97 Uniform and Lipschitz Continuity on Unions
158
98 Uniform and Lipschitz Continuity on Products
159
99 Strong Contractions
160
Summary
162

Summary
18
EXERCISES
19
Distance
21
22 Distances from Points to Sets
22
23 Inequalities for Distances
24
24 Distances to Unions and Intersections
25
25 Isolated Points
26
26 Accumulation Points
28
27 Distances from Sets to Sets
29
28 Nearest Points
30
Summary
32
EXERCISES
33
Boundary
35
32 Sets with Empty Boundary
37
33 Boundary Inclusion
38
34 Boundaries in Subspaces and Superspaces
41
35 Boundaries of Unions and Intersections
42
36 Closure and Interior
43
37 Inclusion of Closures and Interiors
47
38 Closure and Interior of Unions and Intersections
49
Summary
50
Open Closed and Dense Subsets
53
42 Dense Subsets
57
43 Topologies
58
44 Topologies on Subspaces and Superspaces
61
45 Topologies on Product Spaces
62
46 Universal Openness and Universal Closure
64
47 Nests of Closed Subsets
65
Summary
67
Balls
70
52 Using Balls
75
53 Balls in Subspaces and in Products
77
54 Balls in Normed Linear Spaces
78
Summary
81
Convergence
83
62 Limits
85
63 Superior and Inferior Limits of Real Sequences
86
64 Convergence in Subspaces and Superspaces
88
66 Convergence Criteria for Interior and Closure
90
67 Convergence of Subsequences
91
68 Cauchy Sequences
94
69 Cauchy Sequences in Subspaces
97
611 Forcing Convergence of Cauchy Sequences
99
Summary
100
Bounds
103
72 Finite Products of Bounded Sets
105
74 Spaces of Bounded Functions
107
75 Attainment of Bounds
108
76 Convergence and Boundedness
109
77 Uniform and Pointwise Convergence
110
78 Totally Bounded Sets
113
79 Total Boundedness in Subspaces and Superspaces
116
710 Total Boundedness in Product Spaces
117
711 Solution to the NearestPoint Problem
118
712 Subspaces with the NearestPoint Property
121
Summary
123
Continuity
125
82 Limits of Functions
128
83 Global Continuity
130
84 Open and Closed Mappings
135
85 Continuity of Compositions
136
86 Continuity of Restrictions and Extensions
137
87 Continuity on Unions
138
88 Continuity of Mappings into Product Spaces
139
89 Spaces of Continuous Functions
142
810 Convergence as Continuity
143
Summary
144
EXERCISES
145
Uniform Continuity
147
92 Conservation by Uniformly Continuous Maps
150
Completeness
165
102 Criteria for Completeness
166
103 Complete Subsets
168
104 Unions and Intersections of Complete Subsets
169
105 Products of Complete Metric Spaces
170
107 Completeness of the Hausdorff Metric
171
108 Complete Spaces of Functions
173
109 Extending Continuous Functions
176
1010 Banachs FixedPoint Theorem
180
1011 Baires Theorem
183
1012 Completion of a Metric Space
187
Summary
188
EXERCISES
189
Connectedness
191
112 Connected Subsets
193
113 Connectedness and Continuity
194
114 Unions Intersections and Products of Connected Sets
195
115 Connected Components
196
116 Totally Disconnected Metric Spaces
198
118 Pathwise Connectedness
200
119 Polygonal Connectedness
202
Summary
204
Compactness
205
122 Compact Subsets
208
123 Compactness and Continuity
209
124 Unions and Intersections of Compact Subsets
211
125 Compactness of Products
212
127 Local Compactness
213
128 Compact Subsets of Function Spaces
215
129 Paths of Minimum Length
217
1210 FiniteDimensional Normed Linear Spaces
220
1211 A Host of Norms
222
Summary
225
Equivalence
227
132 Uniform Equivalence of Metrics
232
133 Lipschitz Equivalence of Metrics
235
134 The Truth about Conserving Metrics
238
136 Equivalent Metric Spaces
240
Summary
243
Language and Logic
245
A2 Truth of Compound Statements
246
A4 Transitivity of Implication
247
A6 Vacuous Truth
248
A8 Proof by Contraposition
249
A10 Existence
250
Sets
251
B2 Subsets and Supersets
252
B3 Universal Set
253
B5 Ordered Pairs and Relations
254
B6 Totally Ordered Sets
255
B7 Extended Real Numbers
257
B8 Ordered Subsets of the Real Numbers
259
B10 Union Intersection and Difference
260
B11 Unions and Intersections of Arbitrary Collections
261
B12 Functions
262
B13 Restrictions Extensions and Compositions
263
B14 Mappings
264
B15 Chains
266
B16 Equivalence Relations
267
B18 Sequences
268
B19 Infinite Selection
269
B20 Algebraic Structures
272
B21 Isomorphism
275
B22 FiniteDimensional Linear Spaces
277
Solutions
279
List of Symbols
292
Bibliography
295
Index
297
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