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Searcid M. — Metric Spaces
Searcid M. — Metric Spaces

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Название: Metric Spaces

Автор: Searcid M.

Аннотация:

"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.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2006

Количество страниц: 304

Добавлена в каталог: 12.05.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Absolute value      257
Accumulation      28—29
Accumulation, point      28
Addition      272
Algebra      273
Arzela — Ascoli theorem      216 220
Attained, bound      109
Attained, bound criterion      206
Attained, maximum      266
Attained, minimum      266
Axiom of Countable Choice      270
Axiom of Dependent Choice      271
Baire space      185
Baire theorem      185—187 215 290
Balanced set      80
Ball      71—81
Ball in normed linear space      78—81
Ball in product space      77—78
Ball in subspace      77—78
Ball, closed      71
Ball, closed unit      79
Ball, open      71
Ball, open unit      79
Banach, Contraction Principle      180
Banach, Fixed-Point Theorem      181
Barnsley’s fern      183
Basis of linear space      277
BBW criterion      206
Bijective function      265
Binary operation      264
Bolzano–Weierstrass Theorem      122
Bound      103—123
Bound, attained      109 206
Bound, greatest lower      256
Bound, least upper      256
Bound, lower      255
Bound, upper      255
Boundary      35—50
Boundary criterion      191
Boundary in subspace      41—42
Boundary in superspace      41—42
Boundary of intersection      42—43
Boundary of union      42—43
Boundary point      35—37
Boundary, empty      37—38
Bounded function      107—108
Bounded metric space      104
Bounded sequence      109—110
Bounded set      103—104
Bounded set, totally      115
Bounded subset      104
Bounded, totally      113—118
Cantor criterion      206
Cantor intersection theorem      65
Cantor sequence criterion      206
Cantor set      39 44 51 153 168 179 180 198 201 207 281
Cantor set criterion      206
Cardinality      267
Cartesian product      254 260
Cauchy criterion      114 119 166
Cauchy sequence      94 99—100
Cauchy sequence in product      97—98
Cauchy sequence in subspace      97
Chain      266
Class, equivalence      267
Closed      53—67
Closed ball      71
Closed mapping      135
Closed set      38—41 53—57
Closed set criterion      227
Closed union criterion      192
Closed unit ball      79
Closed, algebraically      273
Closure      43—47 90—91
Closure of intersections      49—50
Closure of unions      49—50
Closure, inclusion of      47—48
Closure, universal      64—65
Codomain      264
Collection      251
Combination, linear      277
Compact      205—225
Compact metric space      148
Compact subset      209
Compact, locally      214
Complement      253
complete      165—188 231
Complete metric space      65
Complete order      256
Complete subset      168
Completion      187
Complex function      264
Complex linear space      273
Complex number      254
Component, connected      197
composition      263
Composition, continuity of      157—158
Connected      191—204
Connected component      197
Connected intersection      195—196
Connected metric space      192
Connected product      195—196
Connected subset      193
Connected union      195—196
Connected, pathwise      200
Connected, polygonally      202
Connection, polygonal      202
Conserving metric      14 238
Constant, Lipschitz      154
Constant, sequence      85
containment      251
Continuity      125—144 147—162 170—171 194—195 209—210
Continuity criterion      192
Continuity into product      139—141
Continuity of compositions      136—137 157—158
Continuity of extensions      137—138
Continuity of restrictions      137—138
Continuity on union      138—139
Continuity, global      130—135
Continuity, Lipschitz      154—156 158—160
Continuity, local      125—128
Continuity, uniform      147—150 158—160
Continuous at a point      127
Continuous extension      177—180
Continuous function      131
Continuous functions, space of      142—143
Continuous, uniformly      147
Contraction      160—162
Contraction Principle, Banach      180
Contraction, strong      160
Convergence      83—100
Convergence criterion      119 126 130 206 228
Convergence, pointwise      112
Convergence, uniform      112
Convergent sequence      83—85
Convergent, sequence      99—100
Convex set      79
Coordinate      260
Coordinate set      260 264
Copy, isometric      11
Copy, isomorphic      276
Correspondence, one-to-one      265
Countable      268
Countably infinite      268
Cover      262
Cover, open      206
Criterion, attained bound      206
Criterion, ball      84
Criterion, ball cover      206
Criterion, BBW      206
Criterion, boundary      191
Criterion, BW      119
Criterion, Cantor      206
Criterion, Cantor sequence      206
Criterion, Cantor set      206
Criterion, Cauchy      114 119 166
Criterion, closed set      130 227
Criterion, closed union      192
Criterion, closure      84
Criterion, codomain continuity      228
Criterion, continuity      192
Criterion, convergence      119 126 130 206 228
Criterion, distance      84
Criterion, domain continuity      228
Criterion, epsilon–delta      126
Criterion, epsilon–delta ball      126
Criterion, finite intersection      206
Criterion, global      114
Criterion, identity function      228
Criterion, internal      114
Criterion, local      130
Criterion, nearest-point      118
Criterion, nest      166
Criterion, nested sequence      166
Criterion, open ball      126 130 227
Criterion, open cover      206
Criterion, open set      84 126 130 227
Criterion, open union      191
Criterion, open–closed      191
Criterion, pointlike      118
Criterion, spatial      206
Criterion, universal      166
Criterion, virtual point      166
Curve, Peano      201
Curve, space-filling      201
De Morgan’s theorem      42 43 45 49 62 262 288
Definition, inductive      270
Definition, recursive      270
Degenerate interval      259
Degree of polynomial      275
Dense      57—58
Dense subset      58
Dense, nowhere      69
Denumerable      268
Diameter      21—22
Difference, set      260
Difference, symmetric      19
Differentiable function      156—157
DIMENSION      277—278
Dimension, finite      220—222 277
Dimension, infinite      277
Disconnected      191
Disconnected, totally      198
Discrete metric      4
Discrete metric space      60
Disjoint      261
Disjoint, mutually      261
Distance      2 21—33
Distance between sets      29
Distance from point to set      22—24
Distance to intersections      25—26
Distance to unions      25—26
Distance, inequalities      24—25
Domain      254
Dual      175
Dyadic rational number      58
Element      251
Empty set      251
Empty, boundary      37—38
Endpoint      199
Enumeration      268
Enumerative order      256
Epsilon–delta criterion      126
Equicontinuity      217
EQUIVALENCE      227—243
Equivalence class      267
Equivalence relation      267
Equivalence, Lipschitz      235—237 240
Equivalence, uniform      240
Equivalent metric      227—232
Equivalent norm      238—240
Equivalent space      240—243
Equivalent, Lipschitz      235 240
Equivalent, topologically      229 240
Equivalent, uniformly      232 240
Euclidean metric      3—5
Euclidean product metric      14
Eventually constant sequence      85
Existence      250
Extended natural number      258
Extended real number      258
Extension      263
Extension of metric space      12—13
Extension, continuity of      137—138
Extension, continuous      177—180
Exterior      44
Exterior point      44
Extreme Value Theorem      210
Field      272
Field order isomorphism      276
finite      268
Finite dimension      220—222 277
Finite intersection criterion      206
Finite intersection property      206
Finite sequence      269
Fixed point      180
Fixed-Point Theorem, Banach      181
Fractal      182
Function      262
Function space      173—175
Function, bijective      265
Function, bounded      107—108
Function, complex      264
Function, continuous      131
Function, differentiable      156—157
Function, graph of      264
Function, identity      228 265
Function, injective      265
Function, Lipschitz      154
Function, point      8—10
Function, point evaluation      155 263
Function, pointlike      10
Function, polynomial      275
Function, power      275
Function, real      264
Function, surjective      265
Global continuity      130—135
Graph of function      264
Greatest lower bound      256
Greatest member      255
Half-plane, closed upper      56
Half-plane, open upper      56
Hausdorff metric      106 171—173
Homeomorphic metric spaces      240
Homeomorphism      240
Identity function      228 265
Identity function criterion      228
Image      262 265
Image, inverse      265
Imaginary part      254
inclusion      252
Inductive definition      270
Inequality, triangle      2 3 16
Inferior, limit      87
Infimum      256
Infinite dimension      277
Infinite, countably      268
Injective function      265
INTEGER      253
Integer part      257
1 2 3
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