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Contents 12 Foreword 8 Preface 10 1. Fundamentals of Analysis 14 1.1 Some Elements of Set Theory 14 1.1.1 De Morgan’s laws 14 1.1.2 Equivalence relation 15 1.1.3 Cartesian product and axiom of choice 17 1.2 The Real Numbers 19 1.3 Metric Spaces 22 1.4 Some Useful Concepts 32 1.5 Convergence of Sequences 45 1.6 Convergence of Subsequences 55 2. Continuity and Some Stronger Notions 62 2.1 Basics of Continuity 62 2.2 Continuous Functions: New from Old 71 2.3 Stronger Notions of Continuity 74 2.4 Lipschitz Functions 81 2.5 Isometries between Metric Spaces 89 2.6 Equivalent Metrics 92 3. Complete Metric Spaces 98 3.1 Some Examples 98 3.2 Cauchy Sequences vis-à-vis Total Boundedness 105 3.3 Cantor’s Intersection Theorem 110 3.4 Completion of a Metric Space 114 3.5 Extension Theorems 117 3.6 Baire Category Theorem 127 3.7 Oscillation of a Function 132 4. Compactness 136 4.1 Basic Properties 136 4.2 Equivalent Characterizations of Compactness 141 4.3 Compactness vis-à-vis Continuity 146 4.4 Heine–Borel Theorem 152 5. Weaker Notions of Compactness 158 5.1 Local Compactness 158 5.2 UC Spaces 161 5.3 Cofinally Complete Spaces: A Brief Introduction 169 5.4 Finite Chainability 176 6. Real-Value

Author

Subiman Kundu & Manisha Aggarwal

Language

EN