{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

notes354

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

This preview shows pages 1–6. Sign up to view the full content.

Class Notes for 189-354A. by S. W. Drury Copyright c 2001, by S. W. Drury.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}