An introduction to metric spaces. (2020)

• Record Type: Book
• Further Information: Note: Dhananjay Gopal, Aniruddha Deshmukh, Abhay S. Ranadive, Shubham Yadav.
• Authors: Gopal, Dhananjay; Deshmukh, Aniruddha; Ranadive, Abhay S; Yadav, Shubham
• Contents: Contents Preface ix A Note to the Reader xiii Authors xv 1 Set Theory 1 1.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 The empty set . . . . . . . . . . . . . . . . . . . . . . 4 1.1.2 Operations on sets . . . . . . . . . . . . . . . . . . . . 5 1.1.3 Uniqueness of the empty set . . . . . . . . . . . . . . . 9 1.1.4 Power sets . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.1.5 Cartesian products . . . . . . . . . . . . . . . . . . . . 9 1.2 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.1 Types of relations . . . . . . . . . . . . . . . . . . . . 12 1.2.2 Equivalence relations . . . . . . . . . . . . . . . . . . . 13 1.2.3 Partition of sets . . . . . . . . . . . . . . . . . . . . . 15 1.2.4 Partial order relations . . . . . . . . . . . . . . . . . . 16 1.3 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.3.1 Composition of functions . . . . . . . . . . . . . . . . 24 1.3.2 Inverse of a function . . . . . . . . . . . . . . . . . . . 26 1.3.3 Images of sets under functions . . . . . . . . . . . . . 32 1.3.4 Inverse images of sets under functions . . . . . . . . . 36 1.4 Countability of Sets . . . . . . . . . . . . . . . . . . . . . . . 39 1.4.1 Finite sets . . . . . . . . . . . . . . . . . . . . . . . . . 41 1.4.2 Countable sets . . . . . . . . . . . . . . . . . . . . . . 44 Problem Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Biographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2 Metric Spaces 55 2.1 Review of Real Number System and Absolute Value . . . . . 55 2.2 Young, H¨older, andMinkowski Inequalities . . . . . . . . . . 57 2.3 Notion ofMetric Space . . . . . . . . . . . . . . . . . . . . . 64 2.4 Open Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 2.4.1 Subspace topology . . . . . . . . . . . . . . . . . . . . 96 2.4.2 Product topology . . . . . . . . . . . . . . . . . . . . . 97 2.5 Closed Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 2.6 Interior, Exterior, and Boundary Points . . . . . . . . . . . . 101 2.7 Limit and Cluster Points . . . . . . . . . . . . . . . . . . . . 104 2.8 Bounded Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 110 2.9 Distance Between Sets . . . . . . . . . . . . . . . . . . . . . 112 2.10 EquivalentMetrics . . . . . . . . . . . . . . . . . . . . . . . . 115 Problem Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Biographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . 125 3 Complete Metric Spaces 129 3.1 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 3.1.1 Subsequences . . . . . . . . . . . . . . . . . . . . . . . 130 3.2 Convergence of Sequence . . . . . . . . . . . . . . . . . . . . 131 3.3 CompleteMetric Spaces . . . . . . . . . . . . . . . . . . . . . 139 3.4 Completion ofMetric Spaces . . . . . . . . . . . . . . . . . . 143 3.4.1 Construction of the set Z . . . . . . . . . . . . . . . . 145 3.4.2 Embedding X in Z . . . . . . . . . . . . . . . . . . . . 147 3.4.3 Proving Z is complete . . . . . . . . . . . . . . . . . . 147 3.4.4 Uniqueness of extension up to isometry . . . . . . . . 148 3.5 Baire Category Theorem . . . . . . . . . . . . . . . . . . . . 149 3.5.1 Category of sets . . . . . . . . . . . . . . . . . . . . . 149 3.5.2 Baire category theorem . . . . . . . . . . . . . . . . . 151 3.5.3 Applications of Baire category theorem . . . . . . . . 153 Problem Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Biographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . 158 4 Compact Metric Spaces 161 4.1 Open Cover and Compact Sets . . . . . . . . . . . . . . . . . 161 4.2 General Properties of Compact Sets . . . . . . . . . . . . . . 165 4.3 Sufficient Conditions for Compactness . . . . . . . . . . . . . 169 4.4 Sequential Compactness . . . . . . . . . . . . . . . . . . . . . 172 4.5 Compactness: Characterizations . . . . . . . . . . . . . . . . 174 Problem Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Biographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . 180 5 Connected Spaces 183 5.1 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . 183 5.1.1 Connected subsets . . . . . . . . . . . . . . . . . . . . 185 5.2 Components . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 5.3 Totally Disconnected Spaces . . . . . . . . . . . . . . . . . . 192 Problem Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 6 Continuity 195 6.1 Continuity of Real Valued Functions . . . . . . . . . . . . . . 195 6.2 Continuous Functions in ArbitraryMetric Spaces . . . . . . 197 6.2.1 Equivalent definitions of continuity and other characterizations . . . . . . . . . . . . . . . . . . . . . 202 6.2.2 Results on continuity . . . . . . . . . . . . . . . . . . . 210 6.3 Uniform Continuity . . . . . . . . . . . . . . . . . . . . . . . 217 6.4 Continuous Functions on Compact Spaces . . . . . . . . . . . 224 6.5 Continuous Functions on Connected Spaces . . . . . . . . . . 229 6.5.1 Path connectedness . . . . . . . . . . . . . . . . . . . . 237 6.6 Equicontinuity and Arzela-Ascoli’s Theorem . . . . . . . . . 242 6.7 Open and ClosedMaps . . . . . . . . . . . . . . . . . . . . . 245 6.8 Homeomorphism . . . . . . . . . . . . . . . . . . . . . . . . . 246 Problem Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 Biographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . 252 7 Banach Fixed Point Theorem and Its Applications 255 7.1 Banach Contraction Theorem . . . . . . . . . . . . . . . . . 255 7.2 Applications of Banach Contraction Principle . . . . . . . . . 260 7.2.1 Root finding problem . . . . . . . . . . . . . . . . . . 260 7.2.2 Solution of systemof linear algebraic equations . . . . 261 7.2.3 Picard existence theorem for differential equations . . 264 7.2.4 Solutions of integral equations . . . . . . . . . . . . . 267 7.2.5 Solutions of initial value and boundary value problems . . . . . . . . . . . . . . . . . . . . . . . . . 271 7.2.6 Implicit function theorem . . . . . . . . . . . . . . . . 273 Problem Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 Biographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . 276 Appendix A 277 Bibliography 281 Index 283
• Edition: 1st
• Publisher Details: Boca Raton : Chapman & Hall/CRC
• Publication Date: 2020
• Extent: 1 online resource, illustrations (black and white)
• Subjects: 514.325; Metric spaces
• Languages: English
• ISBNs: 9781000088014; 9781000087994; 9781000088007; 9781003045878
• Related ISBNs: 9780367493486
• Notes: Note: Description based on CIP data; resource not viewed.
• Access Rights: Legal Deposit; Only available on premises controlled by the deposit library and to one user at any one time; The Legal Deposit Libraries (Non-Print Works) Regulations (UK).
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