An Introduction to Metric Spaces / Gopal Dhananjay  

Public ISBD Titre : An Introduction to Metric Spaces Type de document : document électronique Auteurs : Gopal Dhananjay ; Deshmukh Aniruddha ; S. Ranadive Abhay Editeur : Boca Raton : CRC Press Année de publication : 2022 Importance : 1 vol (286 p.) ISBN/ISSN/EAN : 978-1-00-008799-4 Langues : Français (fre) Catégories : Bibliothèque numérique:Mathématique Mots-clés : Metric spaces Index. décimale : 514.3 Topologie des espaces (topologie métrique) Résumé : This book serves as a textbook for an introductory course in metric spaces for undergraduate or graduate students. The goal is to present the basics of metric spaces in a natural and intuitive way and encourage students to think geometrically while actively participating in the learning of this subject. In this book, the authors illustrated the strategy of the proofs of various theorems that motivate readers to complete them on their own. Bits of pertinent history are infused in the text, including brief biographies of some of the central players in the development of metric spaces. The textbook is divided into seven chapters that contain the main materials on metric spaces; namely, introductory concepts, completeness, compactness, connectedness, continuous functions and metric fixed point theorems with applications.Some of the noteworthy features of this book include· Diagrammatic illustrations that encourage readers to think geometrically· Focus on systematic strategy to generate ideas for the proofs of theorems· A wealth of remarks, observations along with a variety of exercises· Historical notes and brief biographies appearing throughout the text Note de contenu : Conten Preface ix A Note to the Reader xiii Authors xv 1 Set Theory 1 1.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.4 Countability of Sets . . . . . . . . . . . . . . . . . . . . . . . 39 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 2.5 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 3.2 Convergence of Sequence . . . . . . . . . . . . . . . . . . . . 131 3.3 CompleteMetric Spaces . . . . . . . . . . . . . . . . . . . . . 139 3.4 Completion ofMetric Spaces . . . . . . . . . . . . . . . . . . 143 3.5 Baire Category Theorem . . . . . . . . . . . . . . . . . . . . 149 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 5.2 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 6.3 Uniform Continuity . . . . . . . . . . . . . . . . . . . . . . . 217 6.4 Continuous Functions on Compact Spaces . . . . . . . . . . . 224 6.5 Continuous Functions on Connected Spaces . . . . . . . . . . 229 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 Problem Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 Biographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . 276 Appendix A 277 Bibliography 281 Index 283 Côte titre : E-Fs/0056 En ligne : https://sciences-courses.univ-setif.dz/login/index.php An Introduction to Metric Spaces [document électronique] / Gopal Dhananjay ; Deshmukh Aniruddha ; S. Ranadive Abhay . - Boca Raton : CRC Press, 2022 . - 1 vol (286 p.). ISBN : 978-1-00-008799-4 Langues : Français (fre) Catégories : Bibliothèque numérique:Mathématique Mots-clés : Metric spaces Index. décimale : 514.3 Topologie des espaces (topologie métrique) Résumé : This book serves as a textbook for an introductory course in metric spaces for undergraduate or graduate students. The goal is to present the basics of metric spaces in a natural and intuitive way and encourage students to think geometrically while actively participating in the learning of this subject. In this book, the authors illustrated the strategy of the proofs of various theorems that motivate readers to complete them on their own. Bits of pertinent history are infused in the text, including brief biographies of some of the central players in the development of metric spaces. The textbook is divided into seven chapters that contain the main materials on metric spaces; namely, introductory concepts, completeness, compactness, connectedness, continuous functions and metric fixed point theorems with applications.Some of the noteworthy features of this book include· Diagrammatic illustrations that encourage readers to think geometrically· Focus on systematic strategy to generate ideas for the proofs of theorems· A wealth of remarks, observations along with a variety of exercises· Historical notes and brief biographies appearing throughout the text Note de contenu : Conten Preface ix A Note to the Reader xiii Authors xv 1 Set Theory 1 1.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.4 Countability of Sets . . . . . . . . . . . . . . . . . . . . . . . 39 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 2.5 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 3.2 Convergence of Sequence . . . . . . . . . . . . . . . . . . . . 131 3.3 CompleteMetric Spaces . . . . . . . . . . . . . . . . . . . . . 139 3.4 Completion ofMetric Spaces . . . . . . . . . . . . . . . . . . 143 3.5 Baire Category Theorem . . . . . . . . . . . . . . . . . . . . 149 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 5.2 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 6.3 Uniform Continuity . . . . . . . . . . . . . . . . . . . . . . . 217 6.4 Continuous Functions on Compact Spaces . . . . . . . . . . . 224 6.5 Continuous Functions on Connected Spaces . . . . . . . . . . 229 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 Problem Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 Biographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . 276 Appendix A 277 Bibliography 281 Index 283 Côte titre : E-Fs/0056 En ligne : https://sciences-courses.univ-setif.dz/login/index.php

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