Bruce K. Driver Undergraduate Analysis Tools February 21, 2013 File:unanal.tex Contents Part I Numbers 1 Natural, integer, and rational Numbers :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 3 1.1 Limits in Q .............................................................................................................................4 1.2 The Problem with Q .....................................................................................................................7 1.3 Peano's arithmetic (Highly Optional) . .9 2 Fields:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 13 2.1 Basic Properties of Fields . 13 2.2 OrderedFields.......................................................................................................................... 15 3 Real Numbers ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 19 3.1 Extended real numbers . 22 3.2 Limsups and Liminfs . 24 3.3 Partitioning the Real Numbers . 29 3.4 The Decimal Representation of a Real Number . 29 3.5 Summary of Key Facts about Real Numbers . 30 3.6 (Optional) Proofs of Theorem 3.6 and Theorem 3.3 . 31 3.7 Supremum and Infiniums of sets . 32 4 Complex Numbers ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 35 4.1 A Matrix Perspective (Optional) . 37 5 Set Operations, Functions, and Counting :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 39 5.1 Set Operations and Functions . 39 5.1.1 Exercises......................................................................................................................... 40 5.2 Cardinality ............................................................................................................................. 41 5.3 FiniteSets.............................................................................................................................. 41 5.4 Countable and Uncountable Sets . 43 4 Contents 5.4.1 Exercises......................................................................................................................... 46 Part II Normed and Metric Spaces 6 Metric Spaces ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 49 6.1 Normed Spaces [Linear Algebra Meets Analysis] . 49 6.1.1 Review of Vector Spaces and Subspaces . 49 6.1.2 Normed Spaces . 50 6.2 Sequences in Metric Spaces . 55 6.3 General Limits and Continuity in Metric Spaces . 58 6.4 Density and Separability . 66 6.5 Test 2: Review Topics . 67 7 Series and Sums in Banach Spaces ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 69 8 More Sums and Sequences ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 85 8.1 Rearrangements . 85 8.2 DoubleSequences ....................................................................................................................... 90 8.3 Iterated Limits . 93 8.4 Double Sums and Continuity of Sums . 94 8.5 Sums of positive functions . 95 8.6 Sums of complex functions . 97 8.7 Iterated sums and the Fubini and Tonelli Theorems . 99 8.8 `p { spaces, Minkowski and H¨olderInequalities . 100 8.9 Exercises............................................................................................................................... 103 8.9.1 Limit Problems . 103 8.9.2 Monotone and Dominated Convergence Theorem Problems . 103 8.9.3 `p Exercises ...................................................................................................................... 105 9 Topological Considerations :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 107 9.1 Closed and Open Sets . 107 9.2 Continuity Revisited . 112 9.3 Metric spaces as topological spaces (Not required Reading!) . 113 9.4 Exercises............................................................................................................................... 114 9.5 Sequential Compactness . ..