Metric Spaces: A Companion to Analysis (Springer Undergraduate Mathematics Series)

Main subject categories: • Mathematical Analysis • Metric Spaces • Baire CategoryMathematics Subject Classification: • 54E35 Metric spaces, metrizability • 54E45 Compact (locally compact) metric spaces • 54E50 Complete metric spaces • 54E52 Baire category, Baire spaces • 46B25 Classical Banach spaces in the general theoryThis textbook presents the theory of Metric Spaces necessary for studying analysis beyond one real variable. Rich in examples, exercises and motivation, it provides a careful and clear exposition at a pace appropriate to the material.The book covers the main topics of metric space theory that the student of analysis is likely to need. Starting with an overview defining the principal examples of metric spaces in analysis (chapter 1), it turns to the basic theory (chapter 2) covering open and closed sets, convergence, completeness and continuity (including a treatment of continuous linear mappings). There is also a brief dive into general topology, showing how metric spaces fit into a wider theory. The following chapter is devoted to proving the completeness of the classical spaces. The text then embarks on a study of spaces with important special properties. Compact spaces, separable spaces, complete spaces and connected spaces each have a chapter devoted to them. A particular feature of the book is the occasional excursion into analysis. Examples include the Mazur–Ulam theorem, Picard’s theorem on existence of solutions to ordinary differential equations, and space filling curves.This text will be useful to all undergraduate students of mathematics, especially those who require metric space concepts for topics such as multivariate analysis, differential equations, complex analysis, functional analysis, and topology. It includes a large number of exercises, varying from routine to challenging. The prerequisites are a first course in real analysis of one real variable, an acquaintance with set theory, and some experience with rigorous proofs. Preface 7 Contents 9 Preliminaries on Sets 14 Basic Relations 14 Basic Operations 14 Writing Predicates 14 Set-Building Rules 15 Relations and Functions 16 Cardinals 17 Other Notions 17 1 Metric Spaces 19 1.1 Metrics 19 1.1.1 Rationale for Metrics 19 1.1.2 Defining Metric Space 20 1.1.3 Exercises 23 1.2 Examples of Metric Spaces 24 1.2.1 Normed Spaces 24 1.2.2 Subspaces 26 1.2.3 Examples; Not Subspaces of Normed Spaces 27 1.2.4 Pseudometrics 27 1.2.5 Cauchy-Schwarz, Hölder, Minkowski 29 1.2.6 Exercises 32 1.3 Cantor's Middle Thirds Set 34 1.3.1 Exercises 37 1.4 The Normed Spaces of Functional Analysis 38 1.4.1 Sequence Spaces 38 1.4.2 Function Spaces 40 1.4.3 Spaces of Continuous Functions 41 1.4.4 Spaces of Integrable Functions 41 1.4.5 Hölder's and Minkowski's Inequalities for Integrals 42 1.4.6 Exercises 43 2 Basic Theory of Metric Spaces 46 2.1 Balls in a Metric Space 46 2.1.1 Limit of a Convergent Sequence 47 2.1.2 Uniqueness of the Limit 48 2.1.3 Neighbourhoods 49 2.1.4 Bounded Sets 49 2.1.5 Completeness; a Key Concept 49 2.1.6 Exercises 52 2.2 Open Sets, and Closed 54 2.2.1 Open Sets 55 2.2.2 Union and Intersection of Open Sets 56 2.2.3 Closed Sets 56 2.2.4 Union and Intersection of Closed Sets 57 2.2.5 Characterisation of Open and Closed Sets by Sequences 58 2.2.6 Interior, Closure and Boundary 59 2.2.7 Limit Points of Sets 60 2.2.8 Characterisation of Closure by Limit Points 60 2.2.9 Subspaces 62 2.2.10 Open and Closed Sets in a Subspace 62 2.2.11 Exercises 64 2.3 Continuous Mappings 69 2.3.1 Defining Continuity 69 2.3.2 New Views of Continuity 70 2.3.3 Limits of Functions 72 2.3.4 Characterising Continuity by Sequences 74 2.3.5 Lipschitz Mappings 74 2.3.6 Examples of Continuous Functions 75 2.3.7 Exercises 76 2.4 Continuity of Linear Mappings 80 2.4.1 Continuity Criterion 81 2.4.2 Operator Norms 83 2.4.3 Exercises 85 2.5 Homeomorphisms and Topological Properties 87 2.5.1 Equivalent Metrics 89 2.5.2 Exercises 90 2.6 Topologies and σ-Algebras 91 2.6.1 Order Topologies 94 2.6.2 Exercises 96 2.6.3 Pointers to Further Study 99 2.7 () Mazur-Ulam 99 2.7.1 Exercises 101 3 Completeness of the Classical Spaces 103 3.1 Coordinate Spaces and Normed Sequence Spaces 103 3.1.1 Completeness of Rn 103 3.1.2 Completeness of p 104 3.1.3 Exercises 106 3.2 Product Spaces 106 3.2.1 Finitely Many Factors 107 3.2.2 Infinitely Many Factors 107 3.2.3 The Space 2N+ and the Cantor Set 109 3.2.4 Subspaces of Complete Spaces 111 3.2.5 Exercises 112 3.3 Spaces of Continuous Functions 115 3.3.1 Uniform Convergence 115 3.3.2 Series in Normed Spaces 117 3.3.3 The Weierstrass M-Test 119 3.3.4 The Spaces C(R) and Cp(R) 119 3.3.5 Exercises 121 3.4 () Rearrangements 124 3.4.1 Vector Series 124 3.4.2 Exercises 127 3.4.3 Pointers to Further Study 127 3.5 () Invertible Operators 128 3.5.1 Fredholm Integral Equation 131 3.5.2 Exercises 132 3.5.3 Pointers to Further Study 134 3.6 () Tietze 134 3.6.1 Formulas for an Extension 137 3.6.2 Exercises 137 3.6.3 Pointers to Further Study 139 4 Compact Spaces 140 4.1 Sequentially Compact Spaces 140 4.1.1 Continuous Functions on Sequentially Compact Spaces 141 4.1.2 Bolzano-Weierstrass in Rn 141 4.1.3 Sequentially Compact Sets in Rn 142 4.1.4 Sequentially Compact Sets in Other Spaces 142 4.1.5 The Space C(M) 143 4.1.6 Exercises 144 4.2 The Correct Definition of Compactness 145 4.2.1 Thoughts About the Definition 146 4.2.2 Compact Spaces and Compact Sets 147 4.2.3 Continuous Functions on Compact Spaces 149 4.2.4 Uniform Continuity 150 4.2.5 Exercises 151 4.3 Equivalence of Compactness and Sequential Compactness 152 4.3.1 Relative Compactness 156 4.3.2 Local Compactness 157 4.3.3 Exercises 158 4.4 Finite Dimensional Normed Vector Spaces 165 4.4.1 Exercises 168 4.5 () Ascoli 169 4.5.1 Peano's Existence Theorem 172 4.5.2 Exercises 175 4.5.3 Pointers to Further Study 177 5 Separable Spaces 178 5.1 Dense Subsets of a Metric Space 178 5.1.1 Defining a Vector-Valued Integral 180 5.1.2 Exercises 183 5.2 Separability 186 5.2.1 Second Countability 187 5.2.2 Exercises 188 5.3 () Weierstrass 191 5.3.1 Exercises 195 5.3.2 Pointers to Further Study 197 5.4 () Stone-Weierstrass 197 5.4.1 Exercises 200 5.4.2 Pointers to Further Study 201 6 Properties of Complete Spaces 202 6.1 Cantor's Nested Intersection Theorem 202 Notes About Cantor's Theorem 203 6.1.1 Categories 203 Thoughts About the Proof 205 6.1.2 Exercises 205 6.2 () Genericity 208 6.2.1 Exercises 211 6.2.2 Pointers to Further Study 214 6.3 () Nowhere Differentiability 214 6.3.1 Exercises 217 6.3.2 Pointers to Further Study 218 6.4 Fixed Points 218 6.4.1 Exercises 219 6.5 () Picard 221 6.5.1 Exercises 224 6.6 () Zeros 224 6.6.1 Exercises 228 6.6.2 Pointers to Further Study 231 6.7 Completion of a Metric Space 231 6.7.1 Other Ways to Complete a Metric Space 233 6.7.2 Exercises 233 7 Connected Spaces 234 7.1 Connectedness 234 7.1.1 Connected Sets 236 7.1.2 Rules for Connected Sets 236 7.1.3 Connected Subsets of R 238 7.1.4 Exercises 238 7.2 Continuous Mappings and Connectedness 239 7.2.1 Continuous Curves 239 7.2.2 Arcwise Connectedness 240 7.2.3 Exiting a Set 241 7.2.4 Exercises 242 7.3 Connected Components 244 7.3.1 Examples of Connected Components 244 7.3.2 Arcwise Connected Components 247 7.3.3 Exercises 247 7.4 () Peano 251 7.4.1 Exercises 251 7.4.2 Pointers to Further Study 252 Afterword 253 Index 255 This textbook presents the theory of Metric Spaces necessary for studying analysis beyond one real variable. Rich in examples, exercises and motivation, it provides a careful and clear exposition at a pace appropriate to the material. The book covers the main topics of metric space theory that the student of analysis is likely to need. Starting with an overview defining the principal examples of metric spaces in analysis (chapter 1), it turns to the basic theory (chapter 2) covering open and closed sets, convergence, completeness and continuity (including a treatment of continuous linear mappings). There is also a brief dive into general topology, showing how metric spaces fit into a wider theory. The following chapter is devoted to proving the completeness of the classical spaces. The text then embarks on a study of spaces with important special properties. Compact spaces, separable spaces, complete spaces and connected spaces each have a chapter devoted to them. A particular feature of the book is the occasional excursion into analysis. Examples include the Mazur–Ulam theorem, Picard’s theorem on existence of solutions to ordinary differential equations, and space filling curves. This text will be useful to all undergraduate students of mathematics, especially those who require metric space concepts for topics such as multivariate analysis, differential equations, complex analysis, functional analysis, and topology. It includes a large number of exercises, varying from routine to challenging. The prerequisites are a first course in real analysis of one real variable, an acquaintance with set theory, and some experience with rigorous proofs.