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

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Class Notes for 189-354A. by S. W. Drury Copyright c 2001, by S. W. Drury. 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 Minkowskis 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 Baires 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...
View Full Document

This note was uploaded on 09/09/2010 for the course MATH MATH 354 taught by Professor Drury during the Fall '06 term at McGill.

Page1 / 212

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online