Metric spaces

Springer, 2006 - 304 Seiten

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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Index

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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

Urheberrecht

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Metric Spaces

Mícheál O'Searcoid
Keine Leseprobe verfügbar - 2006

Häufige Begriffe und Wortgruppen

accumulation point algebraic arbitrary in R+ boundary points bounded functions boundedness Cantor set Cauchy sequence closed subsets closure codomain collection compact subsets complete metric space connected subset conserving metric countable criterion Deﬁnition denote dense subsets diﬀerent discrete metric dist(x domain endowed Euclidean metric Example exists f is continuous ﬁnite ﬁrst follows functions deﬁned G K+ G R+ implies includes a tail inﬁnite injective function interval inverse isolated point isometric copy Lipschitz constant Lipschitz equivalent Lipschitz function locally compact metric subspace metric superspace nearest-point property non-empty closed non-empty metric space non-empty set non-empty subset normed linear space open ball open sets open subset pathwise connected product metric Proof Suppose real numbers satisﬁes sequence xn Show superspace Suppose X,d supremum supremum metric tail of xn Theorem topologically equivalent topology totally bounded triangle inequality usual metric

Bibliografische Informationen

Titel	Metric spaces Springer undergraduate mathematics series, ISSN 1615-2085
Autor	Mícheál Ó Searcóid
Verlag	Springer, 2006
ISBN	1846286271, 9781846286278
Länge	304 Seiten

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Metric spaces

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