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notes354 - Class Notes for 189-354A by S W Drury Copyright...

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Class Notes for 189-354A. by S. W. Drury Copyright c 2001, by S. W. Drury.
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Contents 1 Normed and Metric Spaces 1 1.1 Some Norms on Euclidean Space . . . . . . . . . . . . . . . . . 2 1.2 Inner Product Spaces . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Geometry of Norms . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2 Topology of Metric Spaces 14 2.1 Neighbourhoods and Open Sets . . . . . . . . . . . . . . . . . . 14 2.2 Convergent Sequences . . . . . . . . . . . . . . . . . . . . . . . 16 2.3 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.4 Compositions of Functions . . . . . . . . . . . . . . . . . . . . 25 2.5 Product Spaces and Mappings . . . . . . . . . . . . . . . . . . . 26 2.6 The Diagonal Mapping and Pointwise Combinations . . . . . . . 29 2.7 Interior and Closure . . . . . . . . . . . . . . . . . . . . . . . . 32 2.8 Limits in Metric Spaces . . . . . . . . . . . . . . . . . . . . . . 35 2.9 Distance to a Subset . . . . . . . . . . . . . . . . . . . . . . . . 36 2.10 Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.11 Relative Topologies . . . . . . . . . . . . . . . . . . . . . . . . 40 2.12 Uniform Continuity . . . . . . . . . . . . . . . . . . . . . . . . 44 2.13 Subsequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3 A Metric Space Miscellany 47 3.1 The p -norms on R n . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2 Minkowski’s Inequality and convexity . . . . . . . . . . . . . . . 51 3.3 The sequence spaces p . . . . . . . . . . . . . . . . . . . . . . 54 3.4 Premetrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.5 Operator Norms . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.6 Continuous Linear Forms . . . . . . . . . . . . . . . . . . . . . 62 1
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3.7 Equivalent Metrics . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.8 The Abstract Cantor Set . . . . . . . . . . . . . . . . . . . . . . 66 3.9 The Quotient Norm . . . . . . . . . . . . . . . . . . . . . . . . 67 4 Completeness 70 4.1 Boundedness and Uniform Convergence . . . . . . . . . . . . . 72 4.2 Subsets and Products of Complete Spaces . . . . . . . . . . . . . 76 4.3 Contraction Mappings . . . . . . . . . . . . . . . . . . . . . . . 80 4.4 Extension by Uniform Continuity . . . . . . . . . . . . . . . . . 82 4.5 Completions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.6 Extension of Continuous Functions . . . . . . . . . . . . . . . . 87 4.7 Baire’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.8 Complete Normed Spaces . . . . . . . . . . . . . . . . . . . . . 91 5 Compactness 96 5.1 Compact Subsets . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.2 The Finite Intersection Property . . . . . . . . . . . . . . . . . . 99 5.3 Other Formulations of Compactness . . . . . . . . . . . . . . . 99 5.4 Preservation of Compactness by Continuous Mappings . . . . . . 105 5.5 Compactness and Uniform Continuity . . . . . . . . . . . . . . 108 5.6 Compactness and Uniform Convergence . . . . . . . . . . . . . 111 5.7 Equivalence of Compactness and Sequential Compactness . . . . 112 5.8 Compactness and Completeness . . . . . . . . . . . . . . . . . 115 5.9 Equicontinuous Sets . . . . . . . . . . . . . . . . . . . . . . . . 116 5.10 The Stone–Weierstrass Theorem . . . . . . . . . . . . . . . . . 118 6 Connectedness 128 6.1 Connected Subsets . . . . . . . . . . . . . . . . . . . . . . . . 129 6.2 Connectivity of the Real Line . . . . . . . . . . . . . . . . . . . 131 6.3 Connected Components . . . . . . . . . . . . . . . . . . . . . . 131 6.4 Compactness and Connectedness . . . . . . . . . . . . . . . . . 135 6.5 Preservation of Connectedness by Continuous Mappings . . . . . 136 6.6 Path Connectedness . . . . . . . . . . . . . . . . . . . . . . . . 138 6.7 Separation Theorem for Convex Sets . . . . . . . . . . . . . . . 141 7 The Differential 144 7.1 The Little “o” of the Norm Class . . . . . . . . . . . . . . . . . . 145 7.2 The Differential . . . . . . . . . . . . . . . . . . . . . . . . . . 146 2
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7.3 Derivatives, Differentials and Directional Derivatives . . . . . . . 150 7.4 The Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . 151 7.5 A Lipschitz Type Estimate . . . . . . . . . . . . . . . . . . . . . 154 7.6 One-sided derivatives and limited differentials . . . . . . . . . . 157 7.7 The Differential and Direct Sums . . . . . . . . . . . . . . . . . 158 7.8 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 161 7.9 The Second Differential . . . . . . . . . . . . . . . . . . . . . . 163 7.10 Local Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 8 Integrals and Derivatives 172 8.1 A Riemann type integration theory . . . . . . . . . . . . . . . . 172 8.2 Properties of integrals . . . . . . . . . . . . . . . . . . . . . . . 176 8.3 Taylor’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 181 8.4 Derivatives and Uniform Convergence . . . . . . . . . . . . . . 185 9 The Implicit Function Theorem and its Cousins 189 9.1 Implicit Functions . . . . . . . . . . . . . . . . . . . . . . . . . 190 9.2 Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . 197 9.3 Parametrization of Level Sets . . . . . . . . . . . . . . . . . . . 197 9.4 Existence of Solutions to Ordinary Differential Equations . . . . . 198 3
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1 Normed and Metric Spaces We start by introducing the concept of a norm . This generalization of the absolute value on R (or C ) to the framework of vector spaces is central to modern analysis. The zero element of a vector space V (over R or C ) will be denoted 0 V . For an element v of the vector space V the norm of v (denoted k v k ) is to be thought of as the distance from 0 V to v , or as the “size” of v . In the case of the absolute value on the field of scalars, there is really only one possible candidate, but in vector spaces of more than one dimension a wealth of possibilities arises.
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